Error-Correcting Codes Associated With Generalized Hadamard Matrices Over Groups

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ODU Digital Commons

Mathematics & Statistics Theses & Dissertations Mathematics & Statistics

Winter 1998

Error-Correcting Codes Associated With

Generalized Hadamard Matrices Over Groups

Iem H. Heng

Old Dominion University

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Recommended Citation

Heng, Iem H.. "Error-Correcting Codes Associated With Generalized Hadamard Matrices Over Groups" (1998). Doctor of Philosophy (PhD), dissertation, Mathematics and Statistics, Old Dominion University, DOI: 10.25777/06n5-9c93

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E R R O R C O R R E C T IN G CODES

A SSO C IA T E D W IT H G E N E R A L IZ E D

H A D A M A R D M A T R IC E S O V ER G R O U P S

M.S. April 1995, W estern Michigan University B.S. May 1993, C olum bia University

B.S. May 1991, Providence College

A D issertation S u b m itted to th e Faculty of Old Dominion U niversity in P a rtia l Fulfillment of the

Requirem ent for th e Degree of D O C TO R O F PH ILO SO PH Y

CO M PUTATIONAL AND A P PL IE D MATHEMATICS OLD DOM INION UN IV ERSITY

D ecem ber 1998 by

Iem H. Heng

Approved by:

f l Jo h n M. Dorrepaal (M epiber) Charlie H. Cooke (Director)

HideaJd Kaneko (M ember)

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ER RO R C O R R E C T IN G CODES ASSO CIA TED W ITH G EN ER A LIZED HADAM ARD M A TR IC ES O V E R GROUPS

Iem H. Heng

Old Dom inion University, 1998 D irector: Dr. C harlie H. Cooke

Classical H adam ard m atrices are orthognal m atrices whose elem ents are ± 1 . It is well-known th a t erro r correcting codes having large m inim um distance between codewords can be associated w ith these H adam ard m atrices. Indeed, th e success of early M ars deep-space probes was strongly dependent upon th is com m unication technology.

T h e concept of H ad am ard m atrices with elem ents drawn from an Abelian group is a n a tu ra l generalization of th e concept. For th e case in which the dim ension of th e m atrix is q and th e group consists of th e p -th roots o f unity, these gener alized H adam ard m atrices are called "Butson H adam ard M atrices B H { p , q y . first discovered by A.T. B utson [6].

In th is dissertation it is shown th a t an erro r correcting code whose codewords consist of real num bers in finite Galois field G f ( p ) can be associated in a sim ple way w ith each Butson H adam ard m atrix B H ( p , q ), where p > 0 is a prime num ber. D istance properties of such codes are studied, as well as conditions for the existence of lin ear codes, for which stan d a rd decoding techniques are available.

In th e search for cyclic linear generalized H ad am ard codes, th e concept of an M- invariant infinite sequence whose elem ents are integers in a fin ite field is introduced. Such sequences are periodic of least period, T , an d have the in terestin g p roperty th a t

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a rb itra ry identical rearrangem ents of th e elem ents in each period yields a periodic sequence w ith th e sam e least period. A theorem characterizing such M -invariant sequences leads to discovery of a sim ple and efficient polynom ial m ethod for con stru c tin g generalized H adam ard m atrices whose core is a linear cyclic m atrix and whose row vectors co n stitute a linear cyclic erro r correcting code.

In ad d itio n , th e problem is considered of determ ining p a ra m e te r sequences {£„} for which th e corresponding p o ten tial generalized H adam ard m atrices B H ( p , p t n ) do not exist. By analyzing q u ad ratic D iophantine equations, new m ethods for con stru c tin g such p a ram e te r sequences a re obtained. These results show th e rich num b er th eo retic com plexity of the existence question for generalized H adam ard m a tri ces.

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and my su rro g ate m other B ernadette M ernin

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A C K N O W L E D G M E N T S

I would like to thank m y advisor, Dr. C harlie H. Cooke, who influenced m y decision to work in the area of error correcting codes associated w ith generalized H adam ard m atrices. This is an area w hich is rich in both interesting theories and applications. I am very grateful for m y ad v iso r’s m any helpful insights and th ought ful suggestions. His virtues of patience, encouragem ent, understanding and untiring efforts will long be rem em bered.

I would also like to th a n k Dr. John M. D orrepaal, Dr. Hideaki Kaneko and Dr. Linda L. Vahala for serving on m y co m m ittee. In addition, I am very g ra te ful th a t th e y carefully read my d issertation an d offered thoughtful com m ents and suggestions.

I would like to thank m y m other, m y a u n t and my surrogate m other for th e ir encouragem ent and support during m y sta y in g raduate school. I would also like to th an k my sisters Gucely and Gucemuy, m y brothers Kia, Kimseng and Tong and m y brother-in-law s Kimleng and Neing for encouraging and standing by me.

I would like to express to th e faculty an d staff of the m athem atics departm ent m y appreciation for their o u tstanding su p p o rt. F irst, I would like to th an k B arbara Jeffrey, G ail Tuckelson an d Dr. Rao C h aganty for their assistance w ith Tex and Latex. N ext, I would like to th an k Dr. Jo h n Heinbockel and Dr. John Swetits for th e ir assistance in helping m e to succeed w ith m y graduate studies, by always being available to answer my questions.

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Jam ieson an d Jim T a tte rsa ll who encouraged me to pursue m y goal of o b tain in g a Ph.D . degree in applied m athem atics. W ithout th e ir help in th e past. I would not be w here I now am. I a m very fortunate to have th ese friends.

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Chapter 1 Introduction

1.1 H adam ard M atrices

The m atrix H = H(q, r) is a H adam ard m a trix in th e classical sense provided • H is of dim ension r x r.

• The elem ents of H a re ±1.

• H * H T = H T - H = r l r, where H T is the tra n sp o se of H.

T here is a long history of th e th eo ry of such m atrices, w ith applications in alge braic coding theory, com binatorial design, an d weights and m easures [13], [17], [30], [31], [32] and [33]. Reference [33] contains a n in-d ep th survey of these interesting matrices.

A first generalization of th e concept was discovered by A .T . Butson [5], who defined w hat is called a ‘‘Butson H adam ard m a trix B H ( q , r ) r> having th e following attributes:

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• H = B H ( q , r) has dimension r x r.

• T h e elem ents of H are q-th roots of unity, which lie in the cyclic group G = {1, w , w 2, ..., wq~l : wq = 1}.

• H ■ H “ = H m ■ H = r / r, where is th e conjugate transpose of H. Clearly, th e case q = 2 represents th e classical H adam ard m atrices.

In this dissertation interest is d irected to the case in which q is a prim e n u m  ber greater th an two. A generalization of the concept which encompasses bo th th e classical case and th e complex B utson H adam ard m atrices B H ( q . r ) is th at of gen eralized H adam ard m atrices over general Abelian group G [6]. Such m atrices are denoted by GH{ g, X) or G H ( G , n ) , n = gX, and have th e following attributes: • H = G H ( g , A) has dimension n x n, where n = gX.

• T h e elem ents of H = G H ( g , A) are m em bers of A belian group G, whose order is |G'| = g.

• Homogeneity-Under group operation ®, the vector difference of two arbitrary rows of H satisfies a uniform ity property: Each elem ent of th e group G appears A tim es.

Observe th a t for m ultiplicative groups, the analogy of an elem ent difference is

a — b <— ► ab~l . Clearly, when H — G H ( g , A) has elem ents draw n from the com plex

cylic group Cp = {1, w, w2, ..., wp~l : w p = 1}, then H = B H ( g , g X ) is also B utson H adam ard.

Generalized H adam ard m atrices have been studied by several authors[Brock [4], B utson [5] and [6], Dawson [9], de Launey [10], [11] and [12], D rake [13], Jungnickel [17], S treet [33]]. Brock and S treet use the following definition or some closely related form:

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3

A generalized H adam ard m a trix H = G H{ A s , G ) over th e group G of order s is a As x As m a trix H = [AtJ] whose elements satisfy

(i) hij € G for all 1 < i , j < As,

(” ) H it4i hikh~£ = 12ge G ^9 whenever i ^ j , where th e su m m atio n is in the group ring associated w ith G.

Certainly, if G is the ad d itiv e group from a finite field, J2geG 9 = 0- Thus. Brock’s definition am ounts to a slight generalization of the previous. In th e present work, Brock’s generalization will not be a m ajor concern.

1.2 E x isten ce o f H adam ard M atrices

It is known [22] th a t a necessary condition for existence of th e classical H adam ard m atrices B H ( 2 , r ) is th a t r = 1,2, or a m ultiple of 4k, where k > 0 is an integer. It is a dem o n strated fact [24] th a t BH( ' 2, 4k) exists for each 0 < k < 106. an d a classical H adam ard conjecture is that existence occurs for all integers k > 0. No counter exam ple to this conjecture is presently known.

For prim es p > 2, th e situation is q u ite different. B utson [6] establishes th a t a necessary condition for existence of H = B H ( p > 2. r) is th a t r = pt, w here t > 0 is an integer. T h e condition is also sufficient [5], for th e special case of B H { p > 2 ,2 mpk ), provided 0 < m < k, w here k > 1 an d m are integers.

It has been conjectured fairly recently [3] that B H ( p , p t ) ex ists for all prim es p > 2 w henever t > 0 is an integer. However, instances previously have been

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discovered where this generalized H adam ard con jectu re fails [7]. The m ost recent generalized H adam ard conjecture [10] which ap p ears to have m e rit is th a t B H { p , n) exists only if /„ is H erm itian congruent to n / n, w here n = pt.

As th e circum stances for non-existence of generalized H adam ard m atrices are not com pletely known, one aim of th is d issertatio n will be to obtain p aram eter sequences {<„} for which th e corresponding p o ten tial generalized H adam ard m atrices B H ( p , p t n ) fail to exist.

1.3 S ta tem en t o f P u rp o se

T h e investigations of th is dissertation are directed to the general case of H adam ard m atrices over groups, w ith special consideration given to the question of non-existence, in general, and applications, in p a rtic u la r to th e area of algebraic coding theory.

T h e first focus of in tere st is d ire cted to techniques for constructing p a ram e te r sequences {rt} such th a t th e infinite sequence o f potential generalized H adam ard m atrices B H ( q , rjt) can b e proven non-existent for k € A', where A' is a co u n tab ly in finite set. Techniques exploited consist chiefly of m ethods for proving non-existence of non triv ial integer solutions to homogeneous D iophantine equations

ax2 -f by2 + cz2 = 0. (1.3.1)

T h e second area of em phasis is application o f B utson?s H adam ard m atrice s to th e discipline of algebraic coding theory. It is show n th a t for prim es p > 0 a n error

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correcting code whose codewords consist of real num bers draw n from a finite Galois field G f { p ) can b e associated in a sim ple w ay w ith each existing Butson H adam ard m a trix B H ( p , p t ) . Distance properties of th ese codes, as well as conditions for existence of linear codes, are investigated. By introducing th e concept of an M- invariant periodic infinite sequence whose elem ents are integers in a finite field, an efficient polynom ial m ethod for co n stru ctin g H adam ard m atrices which possess an associated linear cyclic code is m ade posssible.

1.4 B asic L iteratu re S u rvey

A lthough necessary conditions for existence are known, for given integers q and r th e re are no general m ethods for proving non-existence of generalized H adam ard m a trix B H ( q , r ) . However, th e re are several p a ram eter dependent m ethods for investigating specific cases. A .T . Butson [5] d eterm ined th a t for primes p > 2 a necessary condition for the existence of B H { p > 2, r) is th a t r = pt. where t is a positive integer. In addition, he showed th e condition is sufficient for the existence of H adam ard m a trix B H { p > 2 , 2mp*), provided 0 < m < k , where k > 1 an d m are non-negative integers. Existence or non-existence for th e case m > k is generally an open question.

B utson [6] subsequently established th a t th e existence of a generalized H adam ard m a trix H (p, r) is equivalent to th e existence o f H adam ard exponent m atrix, E , whose elem ents are in G alois field Gf { p ) . Here, if H is w ritten in stan d ard form, th e n the first row and first colum n of E a re all zero. W ith o u t th e first row and first column

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of E, the rem aining elem ents c o n stitu te a square su b m atrix , E c, called th e core of H. In order to determ ine a cyclic core E c for th e case w here H(p, pn ) is considered, Butson introduced the concept of constructing a relative difference set. He showed th a t the existence of a relativ e difference set is equivalent to th e existence of cyclic m atrix E c which constitutes a cyclic core for m atrix H ( p , p n ). However, B utson’s m ethod for constructing cyclic m atrix E c by means of a relative difference set is not very practical as m atrix size increases. In the present dissertation, a polynom ial m ethod is introduced for constructing a cyclic m atrix in an efficient m anner.

W hereas Butson studies th e concept of th e existence of H adam ard m atrices, Warwick de Launey [10] focused upon a m eth o d for establishing non-existence. He was able to exploit the n u m b er theoretic ch aracter of th e H adam ard d eterm in a n t to establish non-existence in certain special cases, for groups whose orders are divisible by 3, 5 or 7. In particu lar, th e G H{ 3 ,5 ), G H ( 5,3) and G H ( 15,1) do not exist.

Moreover, de Launey [11] established th a t th e existence of generalized H adam ard matrices G H ( n , G) developed modulo N is equivalent to th e existence of a solution to a certain equation over th e group rin g of G x N over th e integers. H ere, G and N are finite groups of o rd er |G | = g an d \N\ = n, respectively. These m atrices are equivalent to relative difference sets, R D S ( g , n, n ,0 , n/ g ) , m odulo the d irect sum of N and G. By analyzing G H ( q , EA( q) ) developed m odulo EA ( q ) , and G H ( q 2, G) de- vloped m odulo E A ( q 2), w here q is an o d d integer and E A ( q ) denotes th e elem en tary Abelian group of order q, he was able to com e up with som e non-existence results. Indeed, in all b u t 15 of th e 108 cases w ith n < 50, the existence of a G H ( n , EA{ q ) ) ,

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7

developed m odulo E A ( n ) , is either proved or disproved.

F urtherm ore, de Launey [12] determ ined th a t for Abelian groups G , only group H adam ard m atrices of ty p e p5 for Cv x ... x Cp exist. A group H adam ard m atrix H is a generalized H adam ard m a trix whose rows and colum ns form a group. He also defined a group H adam ard m a trix as reducible if there exist two group H adam ard m atrices of sm aller order such th a t H is equivalent to Hi ® H2, denoted as H ~ H\ x /f 2;

otherw ise th e group H adam ard m atrix is said to be irreducible. Two generalized H adam ard m atrices over an Abelian group G are said to be equivalent if one can be o b tained from the o th e r by perm uting the rows an d columns or m ultiplying rows and colum ns by elem ent of G. Through de Launey’s concept of irreducibility, he is able to count what he calls the p(m, t )-H adam ard m atrices.

O n th e other hand, B radley W. Brock [4] generalized th e W itt cancellation law for H erm itian congruence and theorem s of Hall and Ryser [14] to m atrices whose elem ents are members o f a general A belian group. T h e m atrix equation

H • H m = H ' • H = m l n + f i Jn (1.4.2)

is investigated. Here, H m is a conjugate transpose of H of o rd er m, J n is a square m a trix whose elem ents are all l ’s, a n d p is an integer. He obtains a necessary condition for the existence of a nonsingular m atrix H of order m such th a t H • H" = m / n + f i j n. W ith m = n and ft = 0, B rock’s results yield a non-existence theorem for certain generalized H adam ard m atrices, particularly those of th e Butson type.

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m atrices, orthogonal arrays an d F-squares. By com bining known orthogonal arrays associated with generalized H adam ard m atrices, several results of Shrikhande [32] are extended. In addition, established sets of m utually orthogonal F-squares emerge.

M otivated by an exam ple of R ajk u n d lia [28] on b alanced incom plete block de- signs(BIBD ), Jen n ifer Seberry [30] was able to construct m utually orthogonal F- squares from generalized H adam ard m atrice s. First, for th e case where pr and pr — 1 are both prim e powers, with r being an integer, there is a generalized H adam ard m a trix of order pr (pr — 1) w ith elem ents from th e elem entary Abelian group Zp x ... x Z p. This result is then used to produce pr — 1 m utually orthogonal

F-squares F( pr(pr — l) ;p r — 1).

Jennifer S eberry and H. K haraghani [19] have investigated the excess of complex H adam ard m atrices. The excess of a com plex H adam ard m atrix can be obtained as follows: Let H = [/*,_,] be a complex H ad am ard m atrix of order n w ith elem ents 1, -1, i, —i which satisfies H © H" = n /„ , where H~ is th e conjugate transpose of H. Then th e excess of H , denoted as cr(H), is the sum of all entries of H. In m athem atical te rm s, if S ( H) = J^ij th e n cr(H) = |S ( / f ) | is the excess of H. As an application th e re emerges m any real H ad am ard m atrices of large excess and new clsses of Hadam axd m atrices of m axim um excess.

F urtherm ore, w ith W. H olzm ann, H. K haraghani [16] investigated an infinite class of H adam ard m atrices of order n , w here n — 4 is a perfect square, w ith an excess of riy/n — 4. An exhaustive co m p u te r search ind icates th at for n = 40, th e corresponding m a trix constructed has th e m axim um possible excess in its H adam ard equivalence class. T he key to success o f this exhaustive com puter search is s ta r t

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9

ing th e construction using different G olay sequences [34]. T h e re emerges various in teresting classes of H ad am ard m atrices.

Following up w ith th e idea of a G olay sequence, H. Kharaghani [18] went fu rth er to co n stru ct th e class of H adam ard m atrice s, w ith order ( r + 4 n + l)4 n+lm 2, by using

*

th e concept of block G olay sequences. H ere, 2nm is the order of H adam ard m atrix and r is th e length of a Golay sequence, w here m and n are block size a n d length of block Golay sequence, respectively. T h is provides more results on m any regular com plex H adam ard m atrices and H a d a m a rd m atrices of new order.

1.5 O utline o f P ro ced u re

T he purposes of this d issertation a re two-fold: F irst, to determ ine m eth o d s for proving non-existence o f generalized H ad am ard m atrices, and second, to investigate error correcting codes w hich can be associated w ith certain generalized H adam ard m atrices of Butson.

In C h a p te r 2, the su b jec t of generalized H adam ard m atrices over groups is in troduced. T h e aim in th is chapter is to investigate m ethods for establishing th a t certain p aram eter sequences {£„} lead to non-existence of corresponding generalized H adam ard m atrices G H ( s , t n) over group G of order s. T his work com plem ents W arwick de Launey’s approach to non-existence by way of num ber theoretic prop erties o f th e H adam ard d e t e r m in a n t [10]. T h e two investigations, whereas n e ith e r is exhaustive of all possibilities, to g eth er reveal th e rich num ber theoretic com plexity of this existence problem .

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T h e aim of C h a p te r 3 is to investigate erro r correcting codes which can be as sociated with generalized H adam ard m atrices by m eans of th e H adam ard exponent m atrix . In m any cases where D is a difference m a trix , for ap p ro p ria te variable x

H = x ° (1.5.3)

is a generalized H ad am ard m atrix, where th e row vectors o f D form an error cor recting code. B o th linear and nonlinear such codes are possible. It is desirable to determ ine exactly when linear codes occur.

In C hapter 4 th e ta sk is considered of co n stru ctin g generalized H adam ard m a trices whose ex p o n en t m atrix, D , has a cyclic core and whose row vectors co n stitu te a linear cyclic e rro r correcting code. For such cases the ta sk of decoding becomes particularly sim ple. By introducing th e concept of an M -invariant sequence over a finite field, a th eo rem which characterizes such sequences aids in determ ining a sim ple polynomial m eth o d for constructing generalized H adam ard m atrices w ith cyclic core. Having th e polynom ial in h an d also allows one to d eterm in e readily which relative difference set is associated w ith the p artic u la r H adam ard m atrix, although th is concept will not b e explored fully.

In C hapter 5 th e ta sk is to introduce the concept of M -sequences and circulant m atrices. A fter review ing the basic properties a n d theorem s s ta te d in Zierler [37], it

is established th a t every m-sequence is M -invariant. Moreover, because Butson [6]

does not give a clear explanation o r definition o f w hat the cyclic core of a H adam ard m a trix is, a p recursory study of circulant m atrices is accom plished.

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Chapter 2 Criteria for Non-Existence of

Generalized Hadamard Matrices

2.1 In troduction

In this chapter atten tio n is focused upon the question of non-existence for general ized H adam ard m atrices over groups. A well-known conjecture of Brock [4] states th a t generalized H adam ard m atrix G H {p , h) over group G of prim e order p exists only if th e m atrices /„ an d n l n are H erm itian congruent, w here n = ph. T h e CRC H andbook of C om binatorial Design, 1996 edition, docum ents som e p aram eter val ues for which non-existence is known to occur. H ere, m ethods for establishing the lack of solutions to q u ad ratic D iophantine equations are used to prove non-existence o f G H (p ,h ) for various p aram eter sequences. T h e m ethods exploited com plem ent W arwick de Launey’s approach to non-existence via num ber theoretic properties of th e H adam ard d e term in a n t, which is published in J. S ta t. Plann. and

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Infer-ence, V olum e 11, pages 103-110, (1985). N either investigation is exhaustive of all possibilities.

2.2 P relim in ary C on sid eration s

Let C a be th e m ultiplicative group of all com plex s th roots of unity. T he sq u are m a trix H = [/itJ] of order r over Cs is said to be a “Butson Hadamard. m a tr ix ’, briefly a B H ( s ,r ) m atrix , if and only if H H " = r / r . Here. H " is the conjugate tran sp o se of H.

B H { 2, r) m atrices axe referred to sim ply as H ad am ard m atrices (or ± 1 m atrices).

Such m atrices exist only if r = 1, 2 o r else r = 4A:, where k is a positive integer. E xistence has been verified for a t least each an d every k < 106, and the classical H ad am ard conjecture sta te s th at existence occurs for each integer k > 0.

For prim es p > 2. th e situation is q u ite different. A necessary condition for th e existence of B H { p > 2, r) is th a t r = p t , where t is a positive integer. This condition is also sufficient, for the case of B H { p > 2 ,2mpk ), provided 0 < m < k, where k is an in teg er [6].

It has been conjectured [3] th a t B H ( p ,p t) exists, for primes p > 2 and all positive integers t. However, instances have been discovered where this conjecture fails [7]. T h e m ost recent generalized H adam ard conjecture[6] of any m e rit is th a t H (p . n) exists only if /„ is H erm itian congruent to ra/n, w here n = pt.

In th is c h a p te r techniques are explored for proving non-existence of infinite se quences o f p o ten tial B H ( s , r*), k € K , where K is a countably infinite set of positive

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1 3

integers. Sets K are identified for which { B H ( s . r*) : k (E K } = <t>. T h ese techniques consist chiefly of m eth o d s for proving non-existence of nontrivial solutions to homo geneous D iophantine equations

a x 2 4- by2 + c z2 = 0.

2.3 H adam ard M atrices Over Groups

D efin itio n 2 .3 .1 L et (G , G) b e a group o f order g. T (g, k; X)-difference m atrix is

a k x gX m a trix D = (d,j) with entries fro m G, such that fo r each 1 < i < j < k,

the m ultiset

{du 0 d~{1 : 1 < / < gX}

contains every elem ent o f G , A times. When G is Abelian, typically, additive nota tion is used, so that differences du — dji are employed.

Consider th e ad d itiv e group G = {0 ,1 .2 } w ith modulo th ree a rith m etic. Two inequivalent ( 3 ,6; 2)-difference m atrices over G axe

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A = 0 0 0 0 0 0 0 0 1 1 2 2 0 1 0 2 2 1 0 1 2 0 1 2 0 2 2 1 0 1 0 2 1 2 1 0 (2.3.1) and B = 0 0 0 0 0 0 1 2 0 2 0 1 1 0 2 2 1 0 0 2 2 0 1 1 2 2 0 1 1 0 2 0 2 1 0 1 (2.3.2)

D e f in itio n 2 .3 .2 A generalized Hadamard m atrix G H ( g1 A) over group G is a {g.gX; A)-

difference m atrix [7],

A n u m b er of authors have studied theses m atrices [13], [17], [30], [31], [32], and [33]. For a sum m ary of the known m atrices, see Theorem A of S treet [33].

Clearly, b o th difference m atrices A a n d B from (2.3.1) an d (2.3.2), respectively, are generalized H adam ard m atrices G H { 3 ,2 ), each having an associated Butson H adam ard m atrix B H { 3 ,6 ). This association will now be clarified.

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T h eorem 2.3.1 For p rim es p > 2, there exists a generalized Hadamard m atrix BH ( p , p X) over the cyclic group Cp i f and only i f there exists a generalized Hadamard

m atrix GH( p , X) over the additive group Z p = { 0 ,1 ,2 , ...,p — 1}. (+).

A generalization of th is result is s ta te d by Drake [13], whose proof follows from results o f Butson [6]. T h is association will be illu stra ted by exam ple.

Let Cz = { l , x ,x 2}, w here x = e2x,//3 is a prim itive cu b e root of unity. Consider th e Butson H adam ard m atrice s

H = B H ( 3 . 6 ) = x E

where E is one of th e difference m atrices A , B from (2.3.1) and (2.3.2), respectively. T he notation means t h a t m atrix elem ents obey hi3 = x e,J.

By calculation, H H m = 6/; therefore, H is a generalized H adam ard m atrix in th e

classical sense. Also, by calculation H is a G H (3 .2 ) m atrix w ith respect to C3. I'.

T he H adam ard ex p o n en t forms (m atrices .4, B above) have already been c ite d as

G H (3,2) with respect to th e group Z3,© .

The next theorem provides a necessary condition for th e existence of GH[g, X ) over an Abelian group G, w hose order is \G\ = g:

T h eorem 2.3.2 A G H ( g , A) with n = gX odd exists over Abelian group G o f order |G'| = g only if a nontrivial solution in integers ( x , y , z ) exists to the quadratic Diophantine equation

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f o r every order, t, o f a hom om orphic image o f G.

T h e proof of this theorem can be found in Brock [4], an d it is d iscussed in C olbourn an d D initz [7].

C orollary 2.3.1 For p rim es p > 2, and X > 0 an odd integer, BH { p , p X ) exists only i f there are nontrivial solutions in integers (x , y , z) to both equations

z* = p JU 2 + ( - l ) (' - ‘>/W (2 .3 .4 )

and

z 2 = p X x2 + if2. ( 2 .3 .3 )

P ro o f. If G is an A belian group of order p > 2. where p is p rim e, there exist

hom om orphic images of G of orders t = l,p . D

2.4 The Im b ed d in g P rob lem

D e f in itio n 2 .4 .1 Let G be an Abelian group o f order g. with n = gX, where X is a

positive integer. For 0 < k < n, a k x n difference m atrix D o ver the group G is

“completablen i f and only i f there exists a GH( g , X) m atrix having D as its fir s t k rows.

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1 7

T h e H ad am ard imbedding problem concerns the question of w h eth er th e m atrix D can be ex ten d ed by th e process of row addition so as to be com pletable. This problem has been studied variously by B ed er [3], Brock [4], Drake [13] an d others.

D e f in itio n 2 .4 .2 Difference m atrix D o f dim ension k x . n , where n = gX, is “locally

m axim al” (in dim ension) if there is no (k + 1) x n difference m atrix which reduces

to D by deletion o f a single row. I f D is a GH( g , X) , then it is globally m axim al [7J.

It is in tere stin g to note th a t there m ay exist locally m axim al (g , k; A)-difference m atrices for which k < gX, even in cases w here a (g.gX; A)-difference m a trix exists. For g = 2 a n d A = 10, Beder [3] c o n stru cts such (± 1 ) m atrices, characterized by k = 8 ,1 2 ,1 6 .

W ith respect to th e group G = {0, 1,2}, (+ ), by co m p utational m ethods th e present au th o rs have discovered locally m axim al difference m atrices Dkxis with

k = 7,8 (see Tables 1 and 2 on next page). The observation th a t g c d (7 ,15) =

g c d (8 ,15) = 1 appears a stark contrast to w hat m ay be observed in B ed er’s (± 1 ) difference m atrices; namely, in cases w here locally m axim al difference m atrices o f dim ension Dkxn and D nxn sim ultaneously exist, g c d (k ,n ) ^ 1 (for n = 20; k = 8 ,1 2 ,1 6 ).

T his c o n trastin g behaviour leads to th e likely conjecture th a t G H { 3,15) does not exist. A ctually, this has been known for several years. However, following up th is co n jectu re in absence of th is knowledge m otivated the p resen t research on non-existence of certain GH(g, X).

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Tables 1 a n d 2 show th e locally m axim al difference m atrices over group G = { 0 .1.2}, (+ ); previously discussed:

Table 1: (3,7,15)-Difference M atrix

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 2 2 2 2 2 0 0 1 1 1 2 2 0 0 0 1 1 2 2 2 0 0 1 1 2 1 0 2 2 0 2 2 1 1 0 0 0 1 2 2 0 1 1 2 2 1 0 2 0 1 0 1 2 0 2 1 2 0 1 2 1 0 1 2 0 0 1 0 2 2 2 1 2 1 0 0 1 1 0 2 Table 2: ( 3 ,8 J 5 ) - D if f e r e n c e M atrix 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 1 2 1 2 1 2 1 2 0 0 0 0 0 2 1 1 1 1 1 2 0 2 0 2 2 0 0 0 1 2 2 2 0 1 2 1 0 0 1 1 2 0 0 2 1 1 0 2 1 0 2 0 2 _{1 0} 2 1 0 1 2 0 1 2 0 2 0 1 1 2 0 2 1 0 2 0 0 1 2 2 1 1 0 2 2 1 1 0 0 1 0 2 1 2 1 0 2 1 0 0 2 1 2

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19

2.5 Q uadratic D iophan tine E qu ation s

We now consider m ethods for establishing non-existence of nontrivial integer solu tions to th e homogeneous D iophantine equation

a x2 + by2 + cz2 = 0 (2.5.6)

L e m m a 2 .5 .1 I f a and b o f equation(2.5.6) are integers, then the equation

z2 = abx2 ± a y2 (2.5.7)

has nontrivial integer solutions only i f the reduced equation

a l2 = bx2 ± y2 (2.5.8)

has nontrivial integer solutions.

P r o o f . T h e result is obvious. If (x , y , z ) is a solution of (2.5.7), of necessity a\z.

Therefore, let z = a l , where I is a n integer if z is. E

M e t h o d I:

L e g e n d r e ’s T h e o r e m : [29]

Let a, b, c be pairwise relatively prim e integers which are squarefree and n ot all

o f the same algebraic sign. Then equation (2.5.6) has a nontrivial solution in the integers i f and only i f —be, —ac, —ab are quadratic residues o f a, b, c, respectively.

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W arw ick de Launey [10] has approached th e non-existence question for general ized H ad am ard m atrices by m eans of num ber th e o re tic p ro p erties of th e H ad am ard d e te rm in a n t. Basically, he proves th e non-existence of m any generalized H ad am ard

m atrices for groups whose orders a re divisible by 3, 5 o r 7; for exam ple. GH( 15, C15).

G H ( 1 5 , C 3), and G tf (1 5 ,C 5).

T h a t his work is non-exhaustive is evidenced by th e following result:

T h e o r e m 2 .5 .1 For Abelian groups o f order p, and fo r odd p rim es p = ± 3 (m o d 5), G H ( p , 5) does n o t exist.

P r o o f . Consider th e problem o f finding integer solutions to th e equation

z 2 = bpx2 ± p y2, (2.5.9)

w here p = ± 3 (m od 5). This can b e done only if one can find integer solutions of

pq2 = o x2 ± y 2 (2.5.10)

As x 2 = ± 3 (m od 5) has no solutions, by Legendre’s th eo rem n eith er does (2.5.9) or

(2.5.10) have nontrivial integer solutions. a

N o t e . Clearly, Theorem (2.5.1) generalizes som e of de L auney’s results.

2.6 R ecip ro city

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21

D e f in itio n 2 .6 .1 For groups G , H with \G\ = g and \ f f \ = X, potential generalized Hadamard matrices GH( g, X) a n d GH( X, g ) satisfy a reciprocity relation provided both exist or both do not exist.

E x a m p l e . G H (3 ,5 ) and G H ( 5 ,3 ) are reciprocally non-existent, as in each case th e p ertinent reduced equation is of the form

5a2 = 362 + c2.

By Legendre’s theorem , this equation has no nontrivial integer solutions (a. 6. c),

since ± 3 is a quadratic non-residue of o. (T h e concept of q u ad ratic residues is

elab o rated more completely in A ppendix 1.)

By th e sam e approach, the following result can be established:

T h e o r e m 2 .6 .1 Let X be a prim e number. I f ( — l ) 5^ A and ( —l ) i2i A are both

quadratic non-residues o f 5, o r i f ( — 1 ) ^ 5 and (— 1) 5 are both quadratic non

residues o f X, then G H {5, A) and G H {A, 5) constitute a reciprocally non-existent pair.

C o r o l l a r y 2 .6 .1 I f 7 + o k is a prim e number, then G H ( o , 7 + 5 k ) and G H ( 7 + o k , o ) constitute a reciprocally non-existent sequence o f potential generalized Hadamard m atrices.

T h e o r e m 2 .6 .2 L et p = Ak + 3 and q = 4/ + 5 be p rim e numbers, where 2 is a quadratic non-residue o f p. T hen (p. q) is a reciprocal pair.

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t o d eterm in e integer solutions of th e equations

z 2 = pqx2 - py2

an d

z 2 = pqx2 + qy2 ■

E xistence of a nontrivial integer solution of eith e r equation can hap p en only if there exists a nontrivial integer solution (£ , m , n ) for equations of the following type

pH} = q m 2 — n2.

No solution for this equation exists, as

x 2 = 2(m o d p)

has no solution. a

A m ore general m ethod for finding reciprocal pairs employs a result of Euler: E u l e r ’s T h e o r e m : [35]

I f p is an odd prim e which does not divide a, then x 2 = a(m od p) has a solution or no solution according as

a (p i)/2 = i{ m o d p ) (2.6.11)

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2 3

or

q ( p - i)/2 ^ —\(m o d p). (2.6.12)

R e c ip r o c ity T h e o r e m : L et p = 4 k + 3 and a = 4/ + 5 be odd p rim es which sa tisfy E u ler’s condition

a (p-i)/2 s — 1 (mod p).

Then G H ( a , p ) and GH( p, a) constitute a reciprocal non-existent pair o f generalized Hadamard m atrices over groups G. H o f order p, a.

P r o o f . U nder the hypotheses of th e theorem . E u ler’s condition guarantees th e non-existence of nontrivial integer solutions (x, y . z ) to both equations

2 2 2

x = apx - py

and

z2 = a p x2 + ay2,

whose reduced equation is of the form

„n1 2 2

p l ~ a x — y .

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Severed reciprocal pairs are given by Table 3:

Table 3: Reciprocal Pairs

3 5 3 17 3 29 11 13 11 17 11 29 19 29 19 37 19 41 59 61 107 109 M eth o d H :

W hen th e hypotheses of L egendre’s theorem fail, an an aly sis of th e last digit [26] of separate mem bers of equation (2.5.6) is som etim es fruitful. H ere, if x is a nonzero integer, th e last digit of x is den o ted by [x]. For instance, th e last digit of x2 is in th e set

[x2] = { 0 ,1 ,4 ,5 ,6,9 } , and [3x2] = { 0 ,2 ,3 ,5 ,7 ,8 } = [7x2],

[(10A: + l ) ^ 2] = [x2], k > 0 an integer,

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2 5

[5x2] = {0,5},

[9x2] = { 0 ,1 ,4 ,5 ,6,9 }.

These facts are useful in proving some non-existence lem m as below.

L e m m a 2.6 . 1 The equation

z 2 = 3 • 5 - (2k + 1 )x 2 - 3y2 (2.6.13)

where k is a non-negative integer satisfying (2k + 1) ^ 0 (m o d 5), does not possess

a nontrivial solution in integers.

P r o o f . By th e m ethod of contradiction, assum e a nontrivial solution ( x . y . z ) exists, w here (x ,y , x) are non-negative integers. As th e eq u atio n is homogeneous of degree tw o, (x, y, z) is a solution if and only if (tx. iy , t z ) is a solution, where t is an integer. T herefore, it can b e assum ed th a t g c d ( x . y , z ) = 1.

Clearly, z is divisible by 3. If z = 3f, where I is an in teg er, th en equation (2.6.13) reduces to

y2 = 5(2k + l ) x 2 - U 2. (2.6.14)

As the last digit of each in teg er (x2, y2,^2) belongs to th e set L — { 0 ,1 ,4 ,5 ,6,9}, the last digits of 5(2Ar -I- l ) x2 and Zi2 are m em bers of {0,5} and { 0 ,2 ,3 ,5 ,7 ,8}, respectively. For com patibility w ith (2.6.14), th e last d ig it of y2 can only be zero or five; therefore, y = 5m, w here m is an integer.

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Now equation (2.6.14) becomes

U2 = 5(2Ar + l) * 2 - 25m2. (2.6.15)

Therefore, £ = 5p, where p is an integer. E quation (2.6.15) becomes

(2k + l) x2 = 15p2 + 5m2.

Since five does not divide 2k -+- 1, it is necessary th at x = 5q, where q is an integer. T h e conclusions 5|y and 5|x im ply th a t 5|z. As this contradicts gcd(x, y. z) = 1. th e assum ption th a t (2.6.13) has a nontrivial solution in th e integers m ust be false. □

L e m m a 2.6 . 2 The equation

z 2 = 5 • n • (10A: + l ) x2 + 5y2 (2.6.16)

has no nontrivial solution fo r integers k > 0 and n = 1,3, 7.

P r o o f . By the m ethod of contradiction, assum e a nontrivial solution (x , y. z) exists, w here (x , y , z ) are positive integers. As the equation is homogeneous of degree two. (x , y , z ) is a solution if and only if (t x , t y , t z ) is a solution, where t is an integer. Therefore, it can be assum ed th a t g c d ( x . y , z ) = 1.

Clearly, z is divisible by 5 in eq u atio n (2.6.16). C ase 1: n = 1

If z = 5 i , where £ is an integer, th e n equation (2.6.16) reduces to

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2 7

y 2 = 5/ 2 — (lOfc + l)x 2. (2.6.17)

As th e last digit of each integer (x2,^ 2,^ 2) belongs to th e set L = { 0 ,1 .4 ,5 .6 ,9 } ,

th e last digits of and (lOAr + l)x2 a re members of {0,5} and { 0 ,1 .4 ,5 ,6.9 }. respectively. For com patibility w ith (2.6.17), the last dig it of x2 a n d y 2 can only be zero or five; therefore, x = 5m and y = 5p, where m a n d p are integers.

T h e conclusions 5 |y and 5 |x im ply th at 5|x. As this co n tradicts gcd(x*y. z) = 1, the assum ption th a t (2.6.16) has a nontrivial solution in th e integers m u st be false. Case 2: n = 3

If z = 5£, w here I is an integer, then eq u atio n (2.6.16) reduces to

y 2 = U 2 - 3(10fc + l)x 2. (2.6.18)

As th e last digit of each integer (x2, ^ 2,^2) belongs to th e set L = { 0 ,1 .4 ,5 ,6 .9 } ,

th e last digits of o l2 and 3(10fc 4- l)x2 a re members o f {0,5} and { 0 ,2 .3 ,5 , 7.8}. respectively. For com patibility w ith (2.6.18), the last d ig it of y 2 can only be zero or five; therefore, y = 5m, w here m is an integer.

Now equation (2.6.18) becom es

3(10Ar + l)x2 = U 2 - 25m2. (2.6.19)

Since five does not divide 3(10A:+1), it is necessary th a t x = 5p, w here p is an integer. T h e conclusions 5|y and 5 |x im ply th at 5 |z . As this co n trad icts g c d (x , y , z ) = 1, th e assum ption th a t (2.6.16) has a nontrivial solution in th e integers m u st be false.

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Case 3: n = 7

If z = 5£, w here I is an integer, then eq u atio n (2.6.16) reduces to

y 2 = o l2 — 7(10fc + l ) x 2. (2.6.20)

As th e last d ig it of each in te g e r ( x 2, y 2, (?) belongs to th e set L — { 0 ,1 ,4 .5 .6,9 } , th e last digits of b#2 and 7(10A: + l) x2 a re m em bers of {0,5} and { 0 .2 ,3 .5 .7 ,8}, respectively. For co m p atib ility w ith (2.6.20), th e last digit of y 2 can only be zero or five; therefore, y = 5m , w here m is an integer.

Now equation (2.6.20) becom es

7( lO t + l)x2 = b£2 - 25m 2. (2.6.21)

Since five does not divide 7( lOAr+1), it is necessary th a t x = 5p. where p is an integer. T he conclusions 5 |y and 5 |x im p ly th a t b\z. As th is co n trad icts gcd(x. y. r ) = 1, th e assum ption th a t (2.6.16) has a nontrivial so lution in th e integers must be false. □

2.7 Sum m ary

T h e o r e m 2 .7 .1 Several sequences o f p o ten tia l H adam ard matrices over Abelian group G o f order g which do n o t exist are:

1. G # (3,5(2t + 1)), (2k + 1) ^ 0(m o d 5), with k a non-negative integer,

2. G H ( b , n ( l Q k + 1)), f o r n = 1 ,3 ,7 , k non-negative,

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2 9

3. G H( 5 , p ) , where p = ± 3 (m o d 5) is an odd prime,

4 . Reciprocal pairs G H (5 ,7 + 5A:) and G H ( 1 + ok, 5), where 7 + ok is an odd prim e.

C o r o lla r y 2 .7 .1 F o rk a non-negative integer, the following classes o f B H m atrices do not exist:

1. B H { 3 , 15(2A: + 1)), (2k + 1) ^ 0(m od o),

2. B H (b ,b n (\{S k + 1 ) ) , fo r n = 1,3,7,

3. B H ( 5 , b p ) , p = ± 3 (m od 5 ), an odd prim e,

4. Reciprocal pairs B H ( 5,35 + 25k) and B H ( 7 + bk, 35 + 25A:). where 7 + o k is an odd prim e.

T h e following conjecture, which m otivated th is research, appears to gain some sup p o rt from C orollary 2.7.1 and Tables 1 and 2:

C o n j e c t u r e 2 .7 .1 I f fo r 0 < k < gX a locally maxim al (g,k,X)-difference m atrix

with respect to Abelian group G o f order g exists fo r which gcd(k.gX) = 1, then

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Error Correcting Codes

Associated With Complex

Hadamard Matrices Over Groups

3.1 In trod u ction

In this chapter it is shown th a t the row vectors of th e exponent m a trix associated w ith a Butson H adam ard m a trix B H ( p , p t ), p > 0 a prim e num ber, constitute an error correcting code. Some background from algebraic coding th eo ry will therefore now b e introduced.

A block code whose codewords are of length n is a set of vectors of length n whose elem ents axe symbols drawn from som e alphabet. For exam ple, th e alphabet of a p-ary code m ay consist of the sym bols which are elem ents of a finite field Z p = Gf ( p ) , p > 0 a prim e num ber. T hus, any subset K of Z" co n stitu tes a block

30

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31

code of length n. T he code is called a linear group code if an d only if K is a linear subspace. Otherwise, th e code is called nonlinear. K is cyclic if and only if every cyclic p erm u tatio n of th e sym bols in a p a rtic u la r codeword x € K produces another codeword y G K.

T h e H am m ing m etric allows block code K to becom e a m etric space. The distance d ( x , y ) between codewords x and y equals the nu m b er of differences between symbols which occupy corresponding elem en t positions. T h e m inim um distance d ( K) betw een codewords of code AT is a distinguishing fe a tu re of the code.

T h e o r e m 3 .1 .1 (Fundamental Theorem o f Algebraic Coding Theory): A n error correcting code K corrects t errors i f and only i f its m in im u m distance, d ( K) , be tween codewords is greater than 2 t.

The fundam ental param eters which characterize a linear code. A', are (n. k . d). where • n = block length,

• k = num ber of inform ation bits(p er w ord), • d = m inim um distance betw een codew ords.

The inform ation rate R = £ characterizes th e efficiency of a linear code, w here n — k is the n u m b er of redundant bits p er word, w hich is w hat allows error correction to occur. T h e param eter d determ ines capacity to correct errors in transm ission.

T h e purpose of this ch ap ter is to introduce a class of e rro r correcting codes which we call generalized H adam ard codes. Such codes are discovered by analyzing the H ad am ard m atrix of exponents, E , which is associated w ith a com plex (Butson) H ad am ard m atrix, H .

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Recall, H adam ard m a tric e s H(p, q), of index p, are m atrices o f dimension q whose elem ents are p -th roots o f u n ity and whose rows are orthogonal. For the case p = 2, th e elem ents are ± 1 , a n d th e m atrix is referred to as a classical H adam ard m atrix .

For p > 2, th e elem en ts are num bers on th e unit circle, and th e terminology used is th a t of a com plex, o r B utson H adam axd m atrix. References [5], [6], [8] and [32] concern definition, s tru c tu re , properties, an d applications of generalized H adam ard m atrices.

Butson [6] proves t h a t for a fixed p rim e p, a necessary condition for existence of H(p, q) is th a t p div id es q. Thus, in tere st here is directed to com plex H adam ard m atrices H( p , p t ), w here p > 2 is a fixed p rim e and t is a positive integer. W hen such m atrices exist, a real m a trix E ( p , p t ), w hich is called a H adam ard exponent [8]. can be associated w ith H ( p . p t ) . If x is a prim itiv e p-th root of unity, the association is H( p, pt ) = x E(p'pt). T h e notation m eans th at m atrix elem ents are related by kij = x e,>, where i an d j are m atrix indices.

T h e elem ents of th e H adam ard exponent, E , lie in th e G alois field Gf ( p ) . and its row vectors can be view ed as th e codewords of w hat shall be called a generalized H adam ard code. D ep en d in g upon the value of th e non-negative integer t. e ith e r a linear group code or a n o n lin ear code m ay emerge.

3.2 Ternary H adam ard C odes

As a first exam ple of a no n lin ear H adam axd code, th e array Q given below provides th e H adam ard ex p o n en t for a stan d ard form Hadamaxd m a trix H { 3,6) = x Q:

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3 3

T able 4: H ad am ard M atrix H ( 3 ,6) =

* * Q * * * 0 0 0 0 0 0 0 0 1 1 2 2 0 1 0 2 2 1 0 1 2 0 1 2 0 2 2 1 0 1 0 2 1 2 1 0

Clearly, in forming code words by utilizing m atrix row vectors, th e first colum n is superfluous. It may b e seen th a t co d e Q has m inim um distance d(Q) = 4 : therefore, Q corrects up to one e rro r per codew ord when utilized as an erro r correcting code.

T he next exam ple (see Table 5 o n next page) exhibits a linear group code, D , obtained from the H ad am ard ex p o n en t of H{3 ,9 ), again w ritten in stan d ard form.

D has m inim um distance 6; therefore, up to two errors in each tra n sm itted codeword

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Table 5: H adam ard M atrix H [3 .9 ) = * * * * D * * * * 0 0 0 0 0 0 0 0 0 0 1 2 0 1 2 0 1 2 0 2 1 0 2 1 0 2 1 0 0 0 1 1 1 2 2 2 0 1 2 1 2 0 2 0 1 0 2 1 11. 0 2 2 1 0 0 0 0 2 2 2 1 1 1 0 1 2 2 ₀ 1 1 2 0 0 2 1 2 1 0 1 0 2

If th e first ail-zero colum n is om itted, one readily observes th at th e nine row vectors of D = E { 3,9) co n stitu te a ternary linear error correcting code characterized by p aram eters (n. A:, d) — (8,2,6). By augm entation, a n (8,3 ,5 ) code having twenty seven codewords can be o b tained, which has the p u n ctu red f?(3,9) as a subcode.

Sim ilarly, th ere is associated with H {3,27) the ex p o n en t £(3,27) whose corre sponding pu n ctu red linear H adam ard code has p aram eters (26,3,18). This code augm ents to a code which possesses eighty one codewords, an d which has the punc tu re d lineax Hadamaxd code as a subcode.

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3 5

3.3 V ectors Over

Cp

In this section th e problem of establishing th e value, d ( K ), which represents the m inim um Hamming distance betw een the codewords of a generalized H adam ard code, K , is considered.

Let Cp = { l , x , x2, .. ., x p-1} b e th e cyclic group generated by x, w here x =

exp(2icj[p) is a com plex p rim itive p-th root of unity, and p > 2 is a fixed prime.

Further, let A = (xa*), B = (x6,) d e n o te a rb itra ry vectors over C p which a re of length N = p t, where t is a positive integer. Define th e collection of differences between exponents Q = {(a* — 6,)(m od p) : i = 1,2, —iV}, and let nq be th e m u ltip licity of elem ent q of Z p which appears in Q.

P r o p er ty U: Vectors A ,B are sa id to satisfy this property U i f each elem ent q o f Zp appears in Q, exactly t tim es.

The following lem m a is of fu n d am en tal im p o rtan ce in constructing generalized H adam ard codes:

L em m a 3.3.1 (Orthogonality O f Vectors Over Cp) For fixed prim es p, arbitrary vectors A , B whose elements are fro m Cp are orthogonal i f and only i f fo r each element q in Zp, q appears in Q with m ultiplicity t. where N = pt is the length o f A, B, and Q is the collection o f m od p differences between the Hadamard exponents associated with A ,B .

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P ro o f.

S u ffic ie n c y : If Q contains each elem en t q of Zp, t tim es, then th e inner product of A an d B

(A,B) = t - Y , x \

1=0

vanishes, since x is a p-th root of unity. Hence, A an d B are orthogonal.

N e c e s s ity : To th e contrary, suppose n q is not uniform as q varies over Q. If ail nq are non-zero, by using the fact th a t th e sum of all p -th roots of u n ity vanishes, the happy circu m stan ce is cirrived a t w here th e sum involved in ( A . B ) reduces to an integral linear com bination which does n o t involve all p-th roots of unity. Moreover, if any n q = 0, th is circum stance is a lread y present. Because th e coefficients are positive integers, such a linear com bination can not vanish. (Indeed, if p = 3, for any a rb itra ry set of non-zero coefficients th e linear com bination can not vanish, as any two cu b e roots o f unity represent non-collinear vectors in th e plane.) Hence, A

and B are not orthogonal. □

C o m m e n t 1: L em m a 3.3.1 above c an also be inferred from assertions of Butson [6], for which he provides no proof, b u t which he m aintains are clearly valid.

C o m m e n t 2: For any p, P ro p erty U is sufficient for orthogonality. However, if p is not prim e, cases are easily discovered of vectors over Cv which are orthogonal b u t which do n o t satisfy P ro p erty U. T h u s, P ro p erty U is not always necessary for orthogonality.

C o r o l l a r y 3 .3 .1 I f p is a prim e num ber and i f the Hadamard m a trix H {p, pt) exists.

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3 7

the error correcting code, K ( p , p t ) , associated with the corresponding row vectors o f the Hadamard exponent. E( p , p t ) , is characterized by the error protection afforded b y d ( K ) = ( p - l ) t .

P r o o f . In th e m od p difference of any two arbitrary row vectors of th e H adam ard exponent m a trix , th e zero elem ent of Z p appears exactly t times; hence, two code

words differ in (p — 1 )£ sym bols. □

S t a n d a r d F o r m : A ny B u t son's H adam ard m atrix can be transform ed into a H adam ard m a trix for w hich every element o f th e first row and first colum n is unity. In this case, th e first row a n d colum n of th e H adam ard exponent consists of elem ents which are all zero. Som e equivalent of a stan d a rd form m atrix which is obtained by row a n d /o r colum n interchanges is necessary but not sufficient in order to o b tain from E a linear group code, as such codes require presence of the zero vector.

The code words can now be shortened by removing th e first colum n, obtaining what is called a p u n ctu red code, which possesses the sam e level of erro r protection. W hen the code is linear, it can be im bedded in a linear group code having p tim es as many code words, th ro u g h augm entation accomplished by adding appropriate cosets (add, respectively, each elem ent of Z p to each sym bol of each code word, to get a new codew ord).

3.4 E x p o n en t G eneration B y D irect Sum

W hereas th e m atrix Q of Table 4 is tediously obtainable by trial an d error, th e m atrix D o f Table 5 easily follows by use o f a direct su m , em ploying th e m atrix E

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now given,

Table 6: H adam ard Exponent

* £ *

0 0 0 0 1 2 0 2 1

to gether w ith the direct sum below, plus row and colum n interchanges:

L em m a 3.4.1 I f £ = E( p , p ) o f Table 6 is a Hadamard. exponent, then the direct

sum

E ( p . p 2) — ( e 0 + E; i . j = 0,1, ...p — 1)

is a block Hadamard exponent which is also a Hadamard exponent.

Likewise, £ (3 ,2 7 ) may be obtained as th e direct sum of £ (3 ,3 ) and £ (3 .9 ).

3.5 Linear H adam ard C odes

An in itial exploration o f th e properties of th e exponent m atrix associated with a com plex H adam ard m a trix allows identification of a class of generalized H adam ard codes which can be linear or nonlinear, depending upon m atrix dimension N = pt. In this section values o f N are identified which p erm it existence of a sequence of linear H adam ard codes.

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3 9

By definition, code K = K { p , N ) is a linear g roup code if an d only if A" is a linear subspace of 5 = Z VN [1]. T h u s, every linear g roup code can be generated by a finite set Sk of linearly independent vectors in S . If G = G\t is the k x N m atrix w hose row vectors a re th e generators of K , G is called the generator m a trix of K , an d th e param eters ( N , k, d) com pletely ch aracterize K. w here d = d ( K) is th e m inim um H am m ing d istan ce betw een codewords of K . K is called system atic if and only if G = {I k \ B}, w here B is of dimension k x ( N — k).

For prim es p, the sequence of codes characterized by th e rows of the exponent m atrix E ( p , p k ) form a sequence of linear error co rrectin g codes whose codewords are eq u id istan t a t H am m ing distance d ( K ) = pk+l — pk . B utson's results [6] g u ar antee th e existence of such codes: w hereas, linearity has not been established for th e general case. E stablishm ent requires only a d eterm in a tio n th a t th e code space possesses k generating vectors, a p a tte rn which has been verified for k = 2,3 ,4 . T h e following conditions characterize lin ear codes, K = K ( p , N ) :

L em m a 3 .5 .1 A necessary condition that K { p , N ) be linear is that p ^ / N be an integer.

P roof. O therw ise, K is n o t a subgroup of S . □

L em m a 3 .5 .2 I f p is a prim e, a necessary condition that K( p , p t ) be a linear code is that p t be a positive integer power o f p.

P roof. Clearly, if p is a p rim e and N = p t, K (p , N ) m eets the condition of

Error-Correcting Codes Associated With Generalized Hadamard Matrices Over Groups

ODU Digital Commons

Error-Correcting Codes Associated With

Generalized Hadamard Matrices Over Groups

Iem H. Heng

E R R O R C O R R E C T IN G CODES

A SSO C IA T E D W IT H G E N E R A L IZ E D

H A D A M A R D M A T R IC E S O V ER G R O U P S

A C K N O W L E D G M E N T S

Contents

Chapter 1

Introduction

1.1

H adam ard M atrices

1.2

E x isten ce o f H adam ard M atrices

1.3

S ta tem en t o f P u rp o se

1.4

B asic L iteratu re S u rvey

1.5

O utline o f P ro ced u re

Chapter 2

Criteria for Non-Existence of

Generalized Hadamard Matrices

2.1

In troduction

2.2

P relim in ary C on sid eration s

2.3

H adam ard M atrices Over Groups

2.4

The Im b ed d in g P rob lem

2.5 Q uadratic D iophan tine E qu ation s

2.6

R ecip ro city

2.7

Sum m ary

Error Correcting Codes

Associated With Complex

Hadamard Matrices Over Groups

3.1

In trod u ction

3.2

Ternary H adam ard C odes

3.3

V ectors Over

Cp

(A,B) = t - Y , x \

3.4 E x p o n en t G eneration B y D irect Sum

3.5

Linear H adam ard C odes