Code over groups with arbitrary metrics

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Code over groups with arbitrary metrics

Citation for published version (APA):

Bos, A. (1980). Code over groups with arbitrary metrics. (EUT report. WSK, Dept. of Mathematics and Computing Science; Vol. 80-WSK-06). Eindhoven University of Technology.

Document status and date: Published: 01/01/1980

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TECHNISCHE HOGESCHOOL EINDHOVEN NEDERLAND

ONDERAFDELING DER WISKUNDE

EINDHOVEN UNIVERSITY OF TECHNOLOGY THE NETHERLANDS

DEPARTMENT OF MATHEMATICS

Code over groups with arbitrary roetrics

by

A. Bos

T.H. - Report 80-WSK-06

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Code over groups with arbitrary metrics

by

A.Bos

Abstract

A description is given of a method that produces codes over an abelian group with an arbitrary metric. For groups with the Hamming metric the method is known [6], but this paper gives a more general presentation, which makes the construction easier to understand.

O. Introduction

If A is a finite set, a subset of An is called a code over the alphabet A of length n. The Hamming distance between two elements x and y of An is the number of coordinates in which x and y differ. In case 0€ A the weight w(x) of an element x E An is the number of non-zero coordinates of x.

If A

=

{O,l} a code over A is called a binary code. If A is a finite field and C is a linear subspace of An, then C is a linear code. An (n,k,d)-code is a linear code of length n, dimension k and minimum distance d.

Let there be given a binary linear code C with parameters n, k and d and a (N,K,D)-code

r

over the field of 2k elements. Let ~ = (a

1••••,aN), a.

E

GF(2k) be a typical code word of f. There exists a linear

1

Mjection between the elements of GF(2k) and the code words of C. Replacing each coordinate of codeword' ~ by the corresponding binary n-tuple, we obtain a binary vector of length nNe The code obtained in this way has parameters nN, kK and dO and is called a concatenated code [5, Ch 10, § 11].

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- 2

-A generalization of this idea is given by Zinov'ev [6].

In this paper we give a further generalization generating codes over groups with arbitrary metrics.

In the first section we give the definitio~needed,starting from scratch. In section two the main theorem is stated. The third section contains examples in the Hamming , the Lee and the Euclidean metric. In the last section some remarks are made.

1. Some definitions and properties

Let G be an additive group, finite or infinite, with a

metric

d, that is, a mapping d : G x G -+ lR such that for all a,b,c € G

i) d(a,b).?: 0; ii) d(a,b)

o

iff a = b; iii) d(a,b) = d(b,a) and

G -+ lR such that for all

o

iff a = 0; iii) w(a) = we-a) and }v) d(a,b) + d(b,c) .?: d(a,c).

A metric is called

translation invariant

if for all x,a,b E G d(a+x, b+x) = d(a,b).

A

weight function

is a mapping w a,b E G : ·i) w(a) .?: 0; ii) w(a)

iv) w(a) + web) .?: w(a+b).

Clearl~any weight function determines a translation invariant metric and vice versa by d(a,b)

=

web-a) for all a,b E G.

From now on all metrics under consideration are translation invariant. As examples we mention only the Euclidean metric on lRn, the Hamming or trivial metric on an arbitrary group and the Lee metric on ~/m~. The last one is defined by dL(a,b) := min«a-b)mod m, (b-a)mod m) for a,b E

:a-/rnZ.

Let V be a finitely generated free module over G. Each element of V can be represented uniquely by m "coordinates" gl' •.•.• 'gm E G. A metric and a weight function on V, also denoted by d and w respectively, are defined in terms of the metric on G by

m

I

i=l d (g . , h .) and 1 1 w( (g1 ' •••• ,gm» := m

l

i=l w(g. ). 1

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- 3

-A

code

C of length mover G is a subset of V wirh 0 E C and inf

x:Jy x,y E C

d(x,y), the

minimum distance.

Theminimurndistance of the code {a} is CXl by definition.

A code is called a

linear

code if it is a submodule of V.

The

dimension

of a linear code is the number of generators of the coC~. For a linear code we have de = inf w(x).

O:Jx E C For a subset We V and a

E

V we define

d(a,W) := inf d(a,x).

x E

w

Let ~ be a measure on V such that for

we

V with

Iwl

< CXl

~(W)

=

Iw I •

The

centerdensity

0 of a code C is defined by

C

o

:= lim ~(B(O,r) nC)

~(B(O,r» ,

where B(O,r) is the open ball with radius r and center O. Note that for finite G and CC Gm

o

C

The

density

A ofa code C with minimum distance d is defined

C

by

Proof

So for every code we have the

sphere packing condition

A < 1. A code is called

perfect

if equality is attained.

Lemma 1 : If G is finite and CC V is a perfect code, then for all

x E V we have d (x,C) ~ ~<t:

.

Since 0c =

Icil

IGl

m, we must have the equality

Icl.IB(O,~c)

1=

IGlm, thus for all x E V d(x,C) < ~d .0

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Lemma 2

- 4

-Let C be a linear code in V. If Y - x (mod C) then

Proof d(y,C) d(y,C) d(x,C) • inf d(y,a) a E C inf d(x+c,a) a E C inf d (x, a-c) a E C

=

inf d(x,b) b E C

d(x,C), wherey

=

x+c and b a-c, c

E

C.o Corn1_lary 3 : If C is a linear code in V, we can define a metric

on V :=

vic

by w(a) := d(a,C) where a := a+c.

3. Constructions of new codes Theorem 4 : Let C

1 ~ C2 be linear codes in V, C2 := C2/C1 and ~ be an injective homomorfism with ~a E ~ for a E C₂;

r

be a code of length n over C

2 with minimum distance E. Then B:= U (~nx + C~) is a code over G of length

x E

r

nm and with minimum distance D

=

min(E,d C₁). If C₂ is finite, then Os =

Irl

(OC1)n,

Proof Notice that d(~a,C1) w(a) for a E C

2' since ~a E ~ Let x₁,x 2 E

r

be such that xl - x2 = (~l' •.•.•.•'~n) ~ 0 with '(;i E

c

₂• Now we have n

I

d(~iai,c1)

=

i=l > E • Since 0c? 1

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5

-Theorem 5 (Main theorem): Given a tower of linear codes C. (i=O,l, •.• ,k)

1

of length m with minimum distance d., that is_. ₁ CO~C1

=. ....•

='C

k' and injective homomorfisnS<I>i: Ci -+C i such that <I>.a. E a. for a

i E Ci' where C. := C./C. 1Xi=1, ••.• ,k).

1 1 1 1 1

1-Further for each i=l, •.•. ,k a code

r.

_{of length n over Ci and}

1

minimum distance £i is given. Then the new code

.-

u

(r

<I>~x'+Con)

( x_{1 ' ••• , ....k}'· )

E r r

₁_{x •••• x} _k

\

l'

--1

1 1

over G of length nm has minimum distance D = min(d

O'£l •.• '£k).

I f for all i=l, •••• ,k

r

i is a linear code, then so is B

.

k

I f for all i=l, •••• ,k C. is finite, then

_°B

_{= (0} )n

n Ir.]

1 _Co _i=1 ₁

Proof Notice that if Si E C

i for i=r,r+1, •••• ,s then s

d( Is.,

C

r_1) >

des ,

C 1) since

i=r 1 - S

s-with ~ij E Ci. If we suppose that xi=Y_i for i=j+1, ••••,k and x j ~ Yj (j > 1) then k d (

1.

<1>. (xi -Yi ) ' C n ) i=l 1

a

Furthermore j d (

L

<1>. (x1-Y . ), C

~

) • 1=1 1 1 j n j

n

d(

f.

<P.(x.-y.),C~)=

1.

d(

L

<1>.;-. ,CO»

L

d(<I>.a. 'C

j 1) =

i~l 1 1 1 r~l i=l 1 1r - r=l J Jr

-n

I

_{w(a jr)} ~ £j • r=l

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6

-The second statement is obvious.

i f all

I

C .

I

< 00 [J

1.

4. Examples

8

G = {O,l} with the Hamming metric, V = G • Given the tower of 4 codes Co = (8,2 ,4) C C 1 7 8 (8,2 ,2) C C 2 = (8,2 ,1). Then C 1

~

G 3

with twice the Hamming metric and C 2

~

G. . 3 Using f 1 = (4,8 ,4) and 26

E

= (32,2 ,4) over G is 1

f2~ (4,2 ,4) the optimal linear code obtained.

For more examples producing codes over GF(2) with the Hamming metric, see Zinov1ev [6].

ii) G =

m

= V with the Euclidean metric. Given the tower of codes

C =:a' C C =

1.

:ir C C

1.

:a',

o

1 2 2 4

1 1

then C

1 ~ C2 ~ {O,l} with

2

and

4

times the Hamming metric respectively. The codes f 1 = (16,2 11 ,4) and f 2 = (16,2 1 ,1) produce a sphere packing in m16 with highest known center-density 0B = 2 -4

For more examples of sphere packings constructed in this way, see Bos [2] •

iii) Let m > 2 and G = '1f/2m7j = V with the Lee- netric. The codes

o

1 m-l m-J. i m

C

O= (1,2 ,00) C C2 = (1,2,2 )c ••.c Ci = (1,2 ,2 )c ••.c Cm=(1,2 ,1)

- i

form a tower with C, ~ {O,l} with 2 times the Hamming metric.

l.

m Ki m-l The binary i-th order RM-codes f

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i

L

j=l

- 7

-for i=t,2, ••.•• ,m produce the Lee code over ~/ 2m2

B (2m,2K,2m) with K = 2m(

¥

+ 1).

2 m

This improves the (N,N ,N) Lee codes over ~/N~ in [4] for N = 2 •

5. REMARKS

i) If d

1 and d2 are two metrics on the group'G such that for all x,y

E

G d

t (x,y) ~ d2(x,y), we say d2 is a

stronger

metric than dt• If a code C has minimum distance d with respect to metric d

1, and minimum distance d' with respect to a stronger metric d

2, then one has d' > d. Since it is sometimes easier to find the

minimum distance of codes for weaker metrics, this method can be used to give a lower bound for the minimum distance of codes for stronger metrics.

E.g. the Lee-metric on ~/m ~ is stronger than the Hamming metric on 1'/m ~. So every Hamming-code over ~/m ~ is also a Lee-code over ~/m ~ although the estimate for the minimum distance can be rather poor.

i~ Theorem 4 and 5 can also be applied to codes (in a tower) of which

•

one is the union of cosets of the 'other, without them being linear. E.g. Zinov'ev [6] uses the tower of codes

to construct binary codes with minimum distance 24 and 30, although C

3 is the nonlinear extended Nordstrom-Robinson code. The new codes are of course nonlinear.

To produce a good code one must try to find towers of codes that

~~

cannot be refined further and with metrics on the factor groups over which good codes exist.

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iii)

8

-2 Given the binary codes C

O

=

(2,1,d) C C1

=

(2,2 ,1) with d > 2. Then

C

1 ~ ~/4 'I' with the Lee 'metric. So a code

r =

(n,M,d) ov~r

C

₁ produces a (2n,M,d)-code over {0,1}. This is the inverse of a construction noticed by Lee [3]. He remarks that a binary

(2n,M,d)-code produces an (n,M,d)-code over ~/4 ~ with the Lee-metric by substituting 00 by 0, 01 by 1, 10 by 3 and 11 by 2. iv) Following Blokh and Zyablov [1] it seems·to me that for the codes

constructed by theorem 4 and 5, cascade ~odes is a better name than concatenated codes.

REFERENCES

[1] E.L.Blokh & V.Y.Zyablov,

"Coding of generalized cascade codes",

Problemy Peredachi Inform., 1£(1974),45-50.

[2] A.Bos, "Spherepackings in high-dimensional space", (preprint). [3] C.Y.Lee, "Some properties of non-binary error-correcting-codes",

IRE Trans. Inf. Theory, IT-4 (1958), 77-82. [4] Ch.Satyanarayana,

"Lee metric codes over integer residue rings", IEEE IT-25 (1979), 250-254.

[5] F.J.MacWilliams &N.J.A.Sloane,

"The theory of error-correcting codes", North-Holland, Amsterdam 1977.

[6] V.A.Zin~'ev,

"Generalized concatenated codes",

Problemi Peredachi Inform., 12 (1976), 5-15.