Association schemes, codes, and difference sets

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Association Schemes, Codes, and Difference Sets

Thesis by

Carlos H. Salazar-Lazaro

In Partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy

California Institute of Technology Pasadena, California

2007

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c

° 2007

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Acknowledgments

I cannot thank enough my advisor Professor Richard Wilson for his guidance and support throughout my years as a graduate student. I am grateful for his suggestions on the different topics that I have pursued while working on my thesis. I am even more grateful for his patience and kindness.

I thank Cheng-Yeaw Ku, Dinakar Ramakrishnan, and David Wales, the other members of my committee, for their time and patience. I also thank the graduate students who have listened, who have helped me with technical questions, and who have provided invaluable friendships. Specifically, I would like to thank Daniel Katz, Kimball Martin, Jay Bartroff, David Whitehouse, Chris Lee, David Grynkiewicz, Jennifer Johnson, Matthew Gealy, Antonio Hernandez, Rupert Venzke, and many others.

I thank the Caltech mathematics department for supporting me through graduate school. I also thank the math department staff for their invaluable assistance. Specif-ically, I thank Elizabeth Woods, Kathy Carreon, Stacey Croomes, Cherry Galvez, and Seraj Muhammed. I give my special gratitude to Pamela Fong and Sara Mena for being good friends and colleagues. I thank the postdocs whom I befriended, and who helped me in one way or another. Specifically, I thank Vladimir Baranovsky, Cheng-Yeaw Ku, Peter Keevash, Ada Chan, Monica Varizani, Eric Wambach, Eric Rains, David Damanik, and many others.

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Abstract

Adviser: Richard Wilson.

This thesis consists of an introductory chapter and three independent chapters. In the first chapter, we give a brief description of the three independent chapters: abelian

n-adic codes; generalized skew hadamard difference sets; and equivariant incidence structures.

In the second chapter, we introduce n-adic codes, a generalization of the Duadic Codes studied by Pless and Rushanan, and we solve the corresponding existence problem. We introduce n-adic groups, canonical pplitters, and Margarita Codes to generalize the “self-dual” codes of Rushanan and Pless, and we solve the correspond-ing existence problem.

In the third chapter, we consider the generalized skew hadamard difference set (GSHDS) existence problem. We introduce the combinatorial matricesAG,G1, where

G1 = (Z/exp(G)Z)∗ and Gis a group, to reduce the existence problem to an integral

equation. Using a special finite-dimensional algebra or Association Scheme, we show

A2

G,G1 = |G|

p Ifor generalG. With the aid ofAG,G1, we show some necessary conditions for the existence of a GSHDS D in the group (Z/pZ)_×(Z/p2_Z₎2α+1_{, we provide a}

proof of Johnsen’s exponent bound, we provide a proof of Xiang’s exponent bound, and we show a necessary existence condition for general G.

In the fourth chapter, we study the incidence matrices Wt,k(v) of t-subsets of

{1, . . . , v_} vs. k-subsets of _{1, . . . , v_}. Also, given a group G acting on _{1, . . . , v_}, we define analogous incidence matrices Mt,k and M0

t,k of k-subsets’ orbits vs. t

-subsets’ orbits. For general G, we show that Mt,k and M_t,k0 have full rank over Q, we give a bound on the exponent of the Smith Group of Mt,k and M0

t,k, and we give

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the Equivariant Sign Conjecture for the matrices Wt,k(v) using a special basis of the column module ofWt,k consisting of columns ofWt,k; we verify the Equivariant Sign Conjecture for small cases; and we reduce this conjecture to the case v = 2k+t. For the caseG= (Z/nZ), we conjecture thatM_t,k0 has a basis of the column module ofM_t,k0

that consists of columns of M0

t,k. We prove this conjecture for (t, k) = (2,3),(2,4),

and we use these results to calculate the Smith Group of M2,4, M20,4, M2,3, and M20,3

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List of Figures

4.1 Algebraic isomorphisms. . . 166

4.2 Exact sequence integral conditions. . . 171

4.3 Exact sequence integral conditions. . . 172

4.4 Calculating the Frankl rank for the 3-subset _{1,3,4_{} ⊂ {}1, . . . ,7_}. . . . 263

4.5 The orbits of _{1,3,7_} under the action of (Z/7Z). . . 276

4.6 Correspondence between the 3-subsets’ orbits of (Z/7Z) and the parti-tions of 3. . . 284

4.7 f(_{p_}) for k= 4. . . 285

4.8 f(_{p_}) for k= 5. . . 285

4.9 f(_{p_}) for k= 2. . . 286

4.10 f(_{p_}) for k= 3. . . 286

4.11 Relative (2,3)-pod. . . 288

4.12 Projected relative (2,3)-pod. . . 289

4.13 Classes of 3-subsets’ orbits. . . 290

4.14 Relative (2,3)-pod. . . 291

4.16 The Smith Group of M2,3(n). . . 300

4.18 Classes of 4-subsets’ orbits. . . 306

4.19 Relative (2,4)-pod. . . 307

4.21 Relative (2,4)-pod. . . 308

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4.26 Relative (3,4)-pod. . . 340

4.28 Relative (3,4)-pod. . . 342

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Chapter 1

Introduction

1.1

Abelian n-adic Codes

In [21], Pless and Rushanan introduced a family of cyclic binary codes called the Duadic Codes to generalize the cyclic binary Quadratic Residue Codes. Using the language of primitive idempotents1 _in_F

q[G], where Fq is the finite field of q elements

and G is an abelian group, Duadic Codes were introduced as in definition 1.1.1. Definition 1.1.1. Let µ_∈Aut(G), and let e=P_g_∈_Geg[g]be a primitive idempotent in Fq[G]. Define µ·e as Pg∈Geg[σ(g)]. Let e1 =e and e2 =µ·e. If

e1+e2 = 1−

1

|G_|

X

g∈G

[g],

then we say thate1, e2, µforms aDuadic Codeand we call µthesplitting of the code.

Duadic Codes were of special importance since they provided a construction for Finite Projective Planes. This link is sumarized in the following result that is found in [23] in its most general form.

Theorem 1.1.2. LetCbe a duadic code with splitting given byµ₋1 then the minimum

distance for oddlike2 _{code words,} _d

0, satisfies

d2₀₋d0+ 1≥v.

1

An idempotent is primitive, if for every other idempotentf,ef =f e=f 6= 0⇒e=f.

2

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If equality holds then the supports of the minimum-weight oddlike codewords of weight

d0 are the blocks of a projective plane of order (d0−1). Furthermore, d0 is the

mini-mum distance ofC, and the minimum weight codewords are precisely those codewords that are constant on the supports of the blocks of the finite projective plane.

Originally, Duadic Codes were introduced in the cyclic binary case. Since their introduction, there has been various generalizations with respect to:

1. Parity of the characteristic of the field, 2. Number of idempotents used, and 3. The cyclic and noncyclic abelian case.

The following is a history of these generalizations:

1. The family of Duadics in even characteristic, by Rushanan in [22]. These are codes in Fq[G] where G is an arbitrary abelian group, q is even, and 2

idempotents are used in its definition. The existence problem is solved.

2. The family of Duadics in odd characteristic, by Smid in [25]. These are codes in F_q[Z/nZ] where q is odd and 2 idempotents are used in its definition. The existence problem is solved.

3. The family of Triadic Codes, by Pless and Rushanan in [21]. These are codes inFq[Z/nZ] where q is even and 3 idempotents are used in its definition.

The existence problem is solved.

4. The family of Polyadic Codes, by Brualdi and Pless in [4]. These are codes inFq[Z/nZ] andmidempotents are used in its definition. The existence problem

is solved.

5. The family of m-adic Polyadic Codes, by Ling and Xing in [18]. These are codes in Fq[G] whereGis a general abelian group, andm idempotents are used

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We will provide for a different generalization of Duadic Codes that will parallel the family ofm-adic Polyadic Codes of San Ling, and the Duadic Codes of Rushanan. This will be the family of n-adic codes defined by:

Definition 1.1.3. Let n be a prime and G be an abelian group. An n-adic code is a paire, µwheree is an idempotent andµ_∈Aut(G)satisfyingStab_hµi(e) ={µk|µk·e=

e_}=_hµn_i_{, under the action of} _Aut₍_G₎ _{on the primitive idempotents of} _F

q[G], and

he1, . . . , eni = Fq[G], he1i ∩ he2i ∩ · · · ∩ heni = hdg0i

= _h 1

|G_|

X

g∈G

[g]_i,

where ei =µi−1_·_e_.

Remark: The termn-adic, introduced in definition 1.1.3, is not related top-adic numbers.

We will solve the existence problem for n-adic codes as it is given by the next theorem.

Theorem 1.1.4. Let n be a prime, then the following are true:

1. There is an n-adic code inFq[G] if and only if there is an n-adic code in Fq[S]

for every sylow subgroup S of G.

2. There are no n-adic codes in Fq[G] whenG is an n-group.

3. Let G be an s-group where (s, n) = 1. Let ts(k) = |µk| in Aut(Z/sZ). Then,

(a) Let s be an odd prime and assume n divides _ts−1

q(s), then there is an n-adic

code in Fq[G].

(b) Let s be an odd prime, assumen does not divide s−1

tq(s), andG= (Z/sZ) π1_×

(Z/s2_Z₎π2_{× · · · ×}₍Z_/srZ₎πr_{. Then,}

i. Ifs= 1 modulon there is ann-adic code inFq[G]if and only ifπi = 0

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ii. If s₆= 1 modulo n there is an n-adic code in Fq[G]if and only if ts(n)

divides πi for all i.

(c) Let s = 2 be such that (2, n) = 1. Let G= (Z/sZ)π1 _×₍Z_/s2Z₎π2 _{× · · · ×} (Z/sr_Z₎πr_{. There is an} _n_{-adic code in} F

q[G] if and only if tn(2) divides πi

for all i.

We will continue our study ofn-adic codes by providing conditions under whichn -adic codes become Tri-adic Codes, Poly-adic Codes, andm-adic Polyadic Codes. This will put the concept of n-adic codes into perspective with the current generalizations of duadic codes.

We will also provide a generalization of proposition 1.1.2, given as proposition 1.1.5 below, by introducing the concept of n-adic groups for which there will always be a canonical splitter of order n, and by calling the analogous “self-dual” codes Margarita codes.

Theorem 1.1.5. Let e, µl give a Margarita code inFq[G] whereG is ann-adic group

and µl a canonical splitter of order n. Let d0 be the minimum weight of all oddlike

vectors of _he_i. Let w _{∈ h}e_i be an oddlike vector of weight d0. Let w = P_g_∈_Bβg[g]

where B _⊂G is the support of w. Let Be =P_g_∈_B[g] be the characteristic function of the support of w. Define rn,µl,G :G→Z as,

rn,µl,G(g) = |Hg∩(B×. . .×B)|,

where

Hg = {(g1, . . . , gn)∈G× · · · ×G|g1+µl(g2) +· · ·+µln−1(g_n) =g}.

Then,

1. The constant d0 satisfies dn0 −rn,µl,G(1) + 1≥ |G|,

2. If dn

0 −rn,µl,G(1) + 1 = |G|, then:

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(b) The characteristic function of the support of w satisfies,

e

B _∗(µl_·Be)_{∗ · · · ∗}(µ_ln−1_·Be) = (rn,µl,G(1)−1)[1] + X

g∈G

[g].

3. The constant rn,µl,G(1) satisfies rn,µl,G(1) =d0 modulo n.

1.2

Generalized Skew Hadamard Difference Sets

We will assume that the reader is familiar with the theory of Difference Sets as it can be found in Jungnickel’s book in [12], and in Lander’s book in [17]. Also, we will assume thatGis an abelian group, and we will use the additive notation for its group operation.

Skew Hadamard Difference Sets (SHDS) are subsetsDof an abelian groupGsuch that in the group algebra Z[G]:

D(x)D(x−1) = (k₋λ)[0] +λG(x), D(x) +D(x−1) = G(x)₋1.

The following is what is known about the existence problem for SHDSs. This can be found in Chen, Sehgal, and Xiang’s paper in [6].

Theorem 1.2.1. A SHDS has parameters (v, k, λ) = (v,v−1 2 ,

v−3

4 ). Also, if D ⊂ G

is a SHDS, and G is abelian, then:

1. The order of the group v has the form v =p2α+1_{, where} _p_{= 3} _mod ₄_,

2. If χ is any nonprincipal character, then:

χ(D) = −1 +²χ

√ −v

2 ,

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3. If G=Z/pa1Z_×Z_/pa2Z_{× · · · ×}Z_/parZ _where: _a

1 ≥a2 ≥ · · · ≥ar and a1 ≥2,

then:

(a) The exponents a1 and a2 are the same, i.e. , a1 =a2,

(b) The exponents ais satisfy,

a1 ≤

a1+· · ·+ar+ 1

4 .

This is Xiang’s exponent bound.

We introduce the concept of a Generalized Skew Hadamard Difference Set (GSHDS), given by:

Definition 1.2.2. A generalized skew hadamard difference set (GSHDS) is an ele-ment of the group algebra D(x)_∈Z[G] such that:

D(x)D(xno_{) = (}_ko

−λ)[1] +λG(x), D(x) +D(xno_{) =} _G₍_x₎₋_[1]_,

where [1] =x0 _{is the multiplicative unit of} _Z_[_G_]_, _ko ₌_|_D₍_x₎_∩_D₍_x−no₎_|_{, and} _no _{is a}

Non-Quadratic Residue of (Z/exp(G)Z)∗_.

In our study of GSHDSs, we introduce the concept of Quadratic Residue Slices (QRS) defined by:

Definition 1.2.3. Let G be an abelian p-group, and G1 = (Z/exp(G)Z)∗ be the

numerical multipliers of G. Let Ωg1, . . . ,Ωgr be the distinct orbits of the action of G1

on Ge=G_\{0_}. A Quadratic Residue Slice (QRS) is any formal sum of the form:

r

X

i=1

²Ω_giΩgi,

where ²Ω_gi ∈ {±1}.

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Conjecture 1.2.4. If G is an abelian group that affords a GSHDS, then G is ele-mentary abelian.

We prove the analog of proposition 1.2.1 for GSHDS using a unified framework in the language of Quadratic Residue Slices (QRS).

We introduce a combinatorial matrix AG,G1 that is well defined in any abelian

p-group and depends on the choice of a set of representatives of the action of G1 on

G_\{0_}.

Definition 1.2.5. Let G be an abelian p-group with exponent pa1_{. Let} _θ _: _G _→ _Gb

be an noncanonical isomorphism such that θ(g)(g0_{) =} _θ₍_g0₎₍_g₎_{. Let} _Ω

g be an orbit

of G1 = (Z/exp(G)Z)∗ on Ge = G\{0} and Ωχ an orbit of G1 = (Z/exp(G)Z)∗ on

eb

G =Gb_\{χ0}. Then, AG,G1 is defined on the set of orbits of G1 on Ge versus the set

of orbits of G1 on Geb using θ by:

AG,G1(Ωθ(g0),Ωg) =

  

(n_p)o(p_·g) if θ(g0₎₍_g_{) =}_ηn p,

0 else,

where (n_p) is the quadratic residue symbol, o(g) is the order of g in the group G, χ0

is the principal character, ηp is a fixed primitive pth root of unity, and p _· g =p_·g =

g + g + _{· · ·} + g is g added to itself p times.

Using the matrix AG,G1, we introduce the concept of difference coefficients as: Definition 1.2.6. Let d be the vector representation of a QRS D_⊂G, i.e., a vector of _±1. The difference coefficients of D are given by the vector AG,G1d.

We show the following properties of AG,G1: Theorem 1.2.7. The following hold:

1. The incidence matrix AG,G1 satisfies,

A2_G,G₁ = |G|

p Ir,r,

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2. There is a GSHDS D in G if and only if there is a vector d of _±1s such that

AG,G1d=

q

|G|

p d¯for some integral vector d¯.

We proceed to study difference coefficients, and we show integral formulas involv-ing them. In our study of difference coefficients, we prove Xiang’s exponent bound for GSHDS, and we introduce the concept of difference intersection numbers.

Definition 1.2.8. Let D be a QRS in G, and let L _⊂ G be a subgroup with G_L =

{g0

1L, . . . , gs0L}. The difference intersection numbers of D with respect to L inG are

defined as:

νG,L(D, g_i0) = _|D_∩g_i0L_{| − |}D(n0)_∩_g0

iL|.

Using part 1 of theorem 1.2.7 and the structure of Galois Rings, we find necessary conditions for the existence of a GSHDS in the family of groups (Z/pZ)_×(Z/p2_Z₎2α+1_.

We achieve this by first using the multiplicative structure of the Galois RingGR(p2_{, α}₎

to deduce a “Canonical Form” forAH,H1, whereH = (Z/p

2_Z₎2α_{. Using the “Canonical}

Form” of AH,H1, we deduce the following existence conditions:

Theorem 1.2.9. Let G = (Z/pZ) _×(Z/p2_Z₎3_{, and} _D _⊂ _G _{a GSHDS. Let} _H ₌

{0_{} ×}(Z/p2_Z₎3 _⊂ _G _and _L ₌_p_·_H _'₍_Z_/p_Z₎3_{. Let} _D0 ₌_H_∩_D_{, and} _D00 ₌_L_∩_D_.

Let,

1. The vector of _±1s representing D0 _be _d0_.

2. Decompose d0 _by _d0 _{= (}_d0T

0 , d01T, . . . , d0pT2) where d0_i is a (p2+p+ 1)×1 vector of

±1s representing disjoint H1 classes of D0.

3. The vector of difference coefficients of the H1 classes of d0i be die

0

.

Then, there is a ordering of the H1 classes of H and a set of H1-orbit representatives

such that:

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2. The image ofd0 underAL,L1 has formAL,L1d0 =pd¯0 where d¯0 is ap

2₊_p_{+ 1}_×₁

vector of _±1s. Define d¯0 the ”dual” of D00.

3. The image of de0 under AL,L1 has form AL,L1de0 =p ¯

e

d0, for some integral de¯0.

4. The vector d¯e0 of the previous part satisfies de¯0 =Pp

2

i=1di.

5. The vector of _±1s of part 2 satifies pd¯0 =Pp

2

i=1die.

Theorem 1.2.10. Let D be a GSHDS in G= (Z/pZ)_×(Z/p2Z)2β+1, and let K = (Z/pZ)2β_{. Then, there are elements} _{A, B} _∈_Z_[_K_] _{that depend on} _D_{, and there is an}

element L0 ∈Z[K] that depends on the structure of G such that:

1. The elementsA andB satisfy p2β_A₌_χ

0(A)K(x) +L0∗B(−1), whereχ0 is the

principal character of K.

2. The elements A and B satisfy p2β_B ₌ _χ

0(B)K(x) +pL0 ∗A(−1), where χ0 is

the principal character of K.

3. All the coefficients of A are odd, and χ0(A) is odd.

4. All the coefficients of B are odd, and χ0(B) is odd.

5. The principal character of A has value χ0(A) = pβ−1²0b0; where ²0 is ±1, and

b0 is an odd integer.

6. The principal character of B has value χ0(B) = pβ²0a0; where ²0 is ±1 and

equal to the ²0 of part 5, and a0 is an odd integer.

We close the chapter with a proof of Johnsen’s exponent bound using the matrix

AG,G1, and with a necessary existence condition between the Difference Coefficients and the difference intersection numbers of a GSHDS D in a general abelian p-group

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1.3

Equivariant Incidence Structures

We consider the incidence matrices Wt,k defined on the t-subsets vs. k-subsets of {1, . . . , v_} by:

Wt,k(eT, eK) =

  

1 if T _⊂K,

0 else.

The following problems concerning the matrices Wt,k have been solved in the literature:

1. What is the rank of Wt,k? When is Wt,k onto? When is Wt,k injective?

2. How can the kernel of Wt,k be characterized? Do we know a canonical basis for the kernel?

3. Does Wt,k allow for a column Z-basis? That is, is there a subset of the columns of Wt,k that is a basis for the column space of Wt,k overZ? This is the concept of a (t, k)-basis.

4. What is the Smith Normal Form of Wt,k?

5. What is a left inverse of Wt,k whenWt,k is injective? 6. What is a right inverse of Wt,k when Wt,k is surjective?

7. When does an integral vector have an integral preimage under Wt,k?

Questions 1, 5, and 6 have been answered by Kramer in [13, 14, 15]. Questions 3, 4, and 7 have been answered by Wilson in [26]. Question 2 has been answered by Ajoodani-Namini and Khosrovshahi in [1], and by Bier in [3].

We consider these questions when there is an action of a group G. That is, we ask the same questions for Mt,k = (Wt,k)0, and for Mt,k0 = ((Wt,kT )0)T; where Mt,k is

defined by

Mt,k(eΩT, eΩK) = |{K

0 _∈_Ω

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and M0

t,k is defined by

M_t,k0 (eΩT, eΩK) = |{T

0 _∈_Ω

T|T0 ⊂K}|,

where ΩT is an orbit ofG on the t-subsets, and ΩK is an orbit ofG on thek-subsets.

1.3.1

General G

For general G:

1. We show Mt,k and M0

t,k have full rank.

2. We provide a partial answer to the integral preimage problem of Mt,k for

arbi-trary G.

3. We provide some bounds on the exponent of the Smith Group of Mt,k andMt,k0

for arbitrary G.

4. We provide a left inverse of Mt,k when Mt,k is injective.

5. We provide a right inverse of Mt,k when Mt,k is surjective. 6. We provide a left inverse of M0

t,k when Mt,k0 is injective.

7. We provide a right inverse of M_t,k0 when M_t,k0 is surjective. 8. We solve for the Smith Group of M0

t,k provided that for t < k:

(a) The matrix Mt,t0 +1(n) affords a (t, t+ 1)-basis.

(b) A vector z has an integral preimage under Mt,t0 +1 if and only if

¡t+1−i t−i

¢

divides M0

i,tz fori= 1, . . . , t.

1.3.2

Case G

=

Stab

(

Ω

)

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Definition 1.3.1. Let β be a (t, k)-basis of L0

t,k. We say that ² is a trivial positive

equivariant signing for (Lt,k, β), if for every K0 _∈∼β,

Lt,keK0 =

X

K∈β

C(K0, K)²(K)Lt,keK,

we have ²_{∈ {−}1,0,1_} and C(K0_{, K}₎_≥₀_.

We say the signing ² is G-invariant, where G_⊂Sv, whenever²(σ_·K) =²(K) for all σ _∈G and K _∈βt,k.

We consider the action of the group G = Stab(Ω), where Ω is a partition of

{1, . . . , v_}. By introducing the concepts of: monotone families, compatible sets, equiv-ariant compatible sets, and inclusion maps; we give reduction criteria to show for a given G-invariant family βt,k of columns of Wt,k(v):

1. When βt,k is a (t, k)-basis ofWt,k(v). 2. When the set of G-orbits in β, i.e., βG

t,k, is a (t, k)-basis for Mt,k0 .

3. When the set βt,k admits a trivial positive equivariant signing for Wt,k that is

G-invariant. 4. When the set βG

t,k admits a trivial positive equivariant signing for Mt,k0 .

We show the Bier-Frankl (B-F) basis and the Khosorovshahi-Ajoodani (KAj) basis allow a stabilizer group Gof the form Stab(Ωt,k) where

Ωt,k = {1, . . . , v−k−t} ∪ {v−k−t+ 1, v−k−t+ 2} ∪ · · · ∪ {v₋k+t₋1, v₋k+t_{} ∪ {}v₋k+t+ 1, . . . , v_},

for the KAj basis, and

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for the B-F basis. We also show that these bases are equivalent.

Using the reduction results, we prove that these bases admit a trivial positive equivariant signing that is G-invariant for the cases: (1,2),(2,3), by providing a proof; (3,4),(5,6), via a computer program. We also show:

Proposition 1.3.2. Let βt,k be the KAj (t, k)-basis or the B-F (t, k)-basis, then: βt,k(v)admits a trivial positive equivariant signing that isG=Stab(Ωt,k(v))-invariant

for allv _≥k+t+ 1 if and only if βt,k(v) admits a trivially positive equivariant signing

that is G=Stab(Ωt,k(2k+t))-invariant for v = 2k+t.

We propose the following conjecture that we call the “Equivariant Sign Conjec-ture.”

Conjecture 1.3.3. Let βt,k be the KAj (t, k)-basis or the B-F (t, k)-basis, then βt,k

admits a trivial positive equivariant signing that is G=Stab(Ωt,k)-invariant.

1.3.3

Case G

= (

Z

/

nZ

)

For the groupG= (Z/nZ):

1. We solve the integral preimage problem for Mt,k when G = Z/nZ and n is a prime.

2. We calculate the Smith Group of M0

2,3 and of M2,3 for arbitrary n by solving

for a (2,3)-basis.

3. We calculate the Smith Group of M0

2,4 and of M2,4 for arbitrary n by solving

for a (2,4)-basis.

4. We give some partial results for a generating set of the column space of M0

3,4.

We also propose the following conjectures that we call the “(t, k)-basis Conjec-tures.”

Conjecture 1.3.4. (Weak version) Let gcd(tk, n) = 1, and let t < k _≤ n₋k. Then, M0

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Conjecture 1.3.5. (Strong version) Let t < k _≤ n₋ k. Then, M0

t,k admits a

(t, k)-basis.

We close the chapter by showing how the (t, k)-basis conjectures can be used to calculate the Smith Group of M_t,k0 .

Proposition 1.3.6. Let gcd(n,(2(k₋1))!) = 1, and 1 _≤ t < k _≤ n ₋k _≤ n₋t. Assume the(t, k)-basis conjecture holds forn, and for all (i₋1, i) where i= 1, . . . , k. Then, M0

t,k(n) has Smith Group given by:

(Z/

µ

k t

¶

Z)_×(Z/

µ

k₋2

t₋2

¶

Z)r2(n)−1

×(Z/

µ

k₋3

t₋3

¶

Z)r3(n)−r2(n)

× · · ·

×(Z/

µ

k₋(t₋1)

t₋(t₋1)

¶

Z)rt−1(n)−rt−2(n)_,

where ri(n) = ( n i)

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Chapter 2

Abelian n-adic Codes

2.1

Preliminaries

We will consider abelian group codes that are given by ideals in Fq[G] and are also

known as G-codes. We will assume that (q,_|G_|) = 1, where q is a prime power, and that G is abelian. This will make Fq[G] a semisimple algebra and all ideals in Fq[G]

principal (generated by idempotents, unique up to units).

We will assume the natural action of Aut(G) on the idempotents e of G defined for any σ _∈Aut(G) as:

µ_·e = X

g∈G

eg[σ(g)],

where e=P_g_∈_Geg[g].

We will also assume the natural action ofAut(G) onGbdefined for anyσ _∈Aut(G) and χ_∈Gb asσ_·χ=χ_◦σ−1_.

As Fq[G] is semisimple, every G-invariant subspace I1 splits. That is, there is a

G-invariant subspace I2 such that Fq[G] = I1 ⊕I2. In particular, if I1 = hei, then

I2 = h1−ei and Fq[G] = hei ⊕ h1−ei. Hence, Fq[G] = he1i ⊕ · · · ⊕ heri; where

hei_i are ideals that cannot be written as the sum of 2 proper ideals, i.e., irreducible. The idempotents e1, . . . , er are called primitive idempotents. We note that it can be

shown that any idempotent ofFq[G] is the sum of some irreducible idempotents, that

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By considering a splitting field for G, i.e., Fqr where F∗_qr has all exp(G)th roots

of unity, one can show that in Fqr[G]⊇ F_q[G] there are |G| irreducible idempotents

given by the irreducible characters ofGintoF∗

qr; where the idempotent corresponding

to the character χ is given by:

eχ = 1

|G_|

X

g∈G

χ(g−1)[g].

One can show that the Galois group ofFqr overF_q is generated by the Frobenious

Mapx_7→xq_{. We will denote the Frobenious Map by}_σq_{. We note that, one can show}

that ifχ(g) is an irreducible character, thenσq(χ(g)) is another irreducible character. Thus, σq induces an action on the irreducible characters of G.

The following is a well-known result in Character Theory, and it can be found in Isaacs’ book in [10], chapter 9.

Theorem 2.1.1. Lete be the an irreducible idempotent inFq[G], let Fqr be a splitting

field of G. Then, in Fqr[G], e=eχ

1 +· · ·+eχl where χ1, . . . , χl forms an orbit of σq

acting on the irreducible characters of G.

By using theorem 2.1.1, we will proceed to calculate the irreducible idempotents of F_q[G] by introducing the concept of cyclotomic cosets.

Note that σq(χ(g)) = (χ(g))q ₌_χq₍_g_{). Hence, in} _Gb_, _σq _{acts like raising to the} _q_th

power. This mapg _7→gq_{, a numerical multiplier, is clearly in} _Aut₍_G_b_{) as (}_q,_|_G_|_{) = 1.}

By using any noncanonical isomorphism η : G _7→ Gb, where Gb is the group of characters, we can identify anyχ=χg ∈Gb with aη(g). By using this identification,

we can identify the orbit ofχ=χg1 underσq inGb with the orbit of g1 under µq inG. Thus, by theorem 2.1.1, if e is an irreducible idempotent in F_q[G], then there is some orbit of µq acting on G, say Cgo, such that

e = X

g∈Cgo eχg,

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cyclotomic cosets of G.

We will call the irreducible idempotentdg0 =

P

g0_∈_C

1eχ1 =

1

|G|

P

g∈G[g] which

cor-responds to the trivial cyclotomic coset C1 ={1} the trivial irreducible idempotent,

clearly a primitive idempotent in Fq[G].

It can be shown that if X = _{Cgo, . . . , Cgr} are all the distinct µq orbits of G,

then X forms an Association Scheme. That is, if inZ[G] we let C_g0_i =P_g_∈_C

gig, then

there are pk

i,j ∈Z such that:

C_g0_i _∗C_g0_j =

r

X

k=0

pk_i,jC_g0_k.

Thus, the algebra generated by X0 ₌_{_C0

go, . . . , C

0

gr}inFq[G] is finite dimensional

of dimension r+ 1. One can show that this subalgebra is also semisimple and has as idempotents the primitive idempotents of Fq[G]. In particular, this shows that

primitive idempotents, and for that matter any idempotent, are constant on the qth cyclotomic cosets of G. However, there is an easy way of seeing this. Let e be an idempotent, then:

e = eq

= (X

g∈G cg[g])q

= X

g∈G

(cg)q[gq]

= X

g∈G cg[gq] = µq_·e,

which says thatµq is a multiplier ofe. Thus, the coefficients ofeareµqinvariant; that is, they are constant on the qth cyclotomic classes. Note also that Aut(G) induces an action on the primitive idempotents of Fq[G]. We can show this by considering

a primitive idempotent dgi = P

g∈C_gi eχg of Fq[G] ⊆ Fqr[G], where Fqr is a splitting

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µ_·dgi = ( X

g∈C_gi

µ_·eχg)

= X

g∈C_gi

(µ_·eχg),

where, eχg =

1

|G|

P

g0_∈_Gχg((g0)−1)[g0]. Thus,

µ_·eχg =

1

|G_|

X

g0_∈_G

χg((g0)−1)[µ(g0)]

= 1

|G_|

X

g0_∈_G

χg((µ−1(g0))−1)[g0]

= 1

|G_|

X

g0_∈_G

χg(µ−1((g0)−1))[g0]

= 1

|G_|

X

g0_∈_G

(µ_·χg)((g0)−1)[g0].

Define µ∗₍_χ₎₍_g0_{) = (}_µ_·_χ₎₍_g0_{). Note that} _µ∗₍_χ_{) is again an irreducible character} since µ _∈ Aut(G). One can show that µ∗ is an automorphism of the group of irre-ducible characters, i.e.,µ∗ _∈_Aut₍_Gb_{), and that the map}_µ_→_µ∗_{is an anti-isomorphism} fromAut(G) to Aut(Gb).

Clearly, µ_·eχg =eµ∗(χg), and

µ_·dgi = X

g∈C_gi eµ∗₍_χ

g)

= X

g∈µ∗₍_C gi)

eχg,

where µ∗₍_Cg

i) is defined in the following way: since µ∗ ∈ Aut(Gb), and {χg|g ∈ Cgi}

form a cyclotomic coset inGb,µ∗₍_Cg

i) denotes the cyclotomic coset inGcorresponding

to the cyclotomic coset _{µ∗₍_χg₎_|_g _∈_Cg

i}in Gb.

We can be more precise on the definition of µ∗₍_Cg

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η : G _→ Gb. Define ¯µ : G _→ G such that µ∗₍_χg_{) =} _χ

¯

µ(g).1 More precisely, ¯µ is

defined such that µ∗(η(g)) =η(¯µ(g)). Hence, ¯µ(g) =η−1₍_µ∗₍_η₍_g_{))). Clearly,} _µ∗₍_Cg

i)

is ¯µ(Cgi) = Cµ¯(gi) by definition, therefore, µ·dgi =dµ¯(gi).

Thus, µ_·dgi = dµ¯(gi) is a primitive idempotent and this shows that µ acts on

the primitive idempotents the same way ¯µ acts on the cyclotomic cosets. One can show that the map µ _→ µ¯ is an isomorphism of Aut(G) of order two. So, given

σ _∈Aut(G), there is a unique τ _∈Aut(G) such that σ= ¯τ. Also, the action ofµ on the primitive idempotents is the same as the action of ¯µon the cyclotomic cosets via the correspondence established by η.

We will define X =_{dg0, dg1, . . . , dgr} as the set of all the primitive idempotents

of Fq[G]. Also, for given Y ⊂ X, we will define eY = P_d_g_∈_Y dg as the idempotent

of Fq[G] that corresponds to Y. We note that the previous definition is exhaustive.

That is, given an idempotent e of Fq[G] there is Y ⊂X such thate =eY.

Using the previous notation, one can show:

eX = 1, e_∅ = 0,

eX1 ∗eX2 = eX1∩X2, (2.1)

heX1, eX2i = heX1∪X2i, (2.2)

µ_·eX1 = eµ(X1), (2.3)

where X1 ⊂X, X2 ⊂X, andµ(X1) ={µ·dg =dµ¯(g)|dg ∈X1}.

2.2

Abelian n-adic Codes

Definition 2.2.1. We will assume that n is a prime. An n-adic code is a pair e, µ

where e is an idempotent and µ _∈ Aut(G) satisfying Stab_hµi(e) = {µk|µk·e = e} =

1

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hµn_i _and

he1, . . . , eni = Fq[G], (2.4)

he1i ∩ he2i ∩ · · · ∩ heni = hdg0i (2.5) = _h 1

|G_|

X

g∈G

[g]_i,

where ei =µi−1_·_e_{. We will call} _µ _{the splitter of the code.}

Definition 2.2.2. We will say that ann-adic codeµ, eis“reduced”if_hei_i∩hej_i=dg0

for all i, j.

Examples of n-adic codes are given by the Q-codes of Rushanan in [22], which are

n-adic codes where n = 2.

The main objective of this section is to solve the existence problem for abelian

n-adic codes in F_q[G]. This is is summarized by theorem 2.2.3. Theorem 2.2.3. The following are true:

1. There is an n-adic code inF_q[G] if and only if there is an n-adic code in F_q[S]

for every sylow subgroup S of G.

2. There are no n-adic codes in Fq[G] whenG is an n-group,

3. Let G be an s-group where (s, n) = 1. Let ts(k) = _|µk_| in Aut(Z/sZ). Then, (a) Let s be an odd prime. Assumen divides s−1

tq(s), then there is ann-adic code

in Fq[G].

(b) Lets be an odd prime. Assumendoes not divide _ts_q−₍_s1₎, and G= (Z/sZ)π1_× (Z/s2_Z₎π2_{× · · · ×}₍Z_/srZ₎πr_{. Then,}

i. Ifs= 1 modulon there is ann-adic code inFq[G]if and only ifπi = 0

modulo n for all i.

ii. If s₆= 1 modulo n there is an n-adic code in Fq[G]if and only if ts(n)

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(c) Let s = 2 be such that (2, n) = 1. Let G= (Z/sZ)π1 _×₍Z_/s2Z₎π2 _{× · · · ×} (Z/sr_Z₎πr_{. There is an} _n_{-adic code in} F

q[G] if and only if tn(2) divides πi

for all i.

We proceed to show existence conditions for n-adic codes. The next subsections will take care of the individual cases of theorem 2.2.3.

2.2.1

Existence Criteria

We will show criteria for existence of n-adic codes that will be used in the proof of theorem 2.2.3.

A direct consequence of then-adic code definition is that the order of the splitter

|µ_|inAut(G) is divisible by n. We can show this by contradiction. If _|µ_|=n_∗n0₊_l where l_{∈ {}1, . . . , n₋1_}, then

e = µ|µ|_·e

= µn∗n0+l_·e

= µl_·e,

which contradicts the choice of n as the smallest power of µ such thatµk_·_e₌_e_.

Using equations (2.1) and (2.3), and the n-adic code definition, we can force set conditions. Let e = eY for some Y ⊂ X where e, µ give an n-adic code, then ei =µi−1_·_e₌_e

µi−1₍_Y₎. Equations (2.1), (2.3), and (2.5) give the following:

dg0 = e1∗ · · · ∗en

= eY ∗eµ(Y)∗ · · · ∗eµn−1₍_Y₎ = eY∩µ(Y)∩···∩µn−1₍_Y₎.

Thus,

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Similarly, equations (2.2) and (2.4) give:

Y _∪µ(Y)_{∪ · · · ∪}µn−1(Y) = X. (2.7)

We can use the previous set conditions to deduce properties of the splitter µ. Corollary 2.2.4. Let, e, µ give an n-adic code in F_q[G], then:

1. If there is a primitive idempotent d such that d _∗ µ = d then d = dg0 =

1

|G|

P

g∈G[g].

2. If there is a cyclotomic coset such that µ¯(Cg) = Cg then Cg =_{1_}.

Proof. Both conclusions are equivalent, as the action of µ on the primitive idempo-tents is the same as the action of ¯µ on the cyclotomic cosets and dg0 corresponds to

C1 ={1}. Only the first statement will be shown.

Let d be a primitive idempotent; then Equation (2.7) says that d _∈ µk₍_Y_{) for}

some k. That is, d = µk _· _d0 _{for some} _d0 _∈ _Y_{. Suppose} _µ_· _d ₌ _d_{. Note that}

µk+l_·_d0 ₌_µl_·₍_µk_·_d0_{) =} _µl_·_d ₌_d _{which shows that} _d _∈ _µk+l₍_Y_{) for all} _l_{. That is,} d_∈µl₍_Y_{) for all} _l_{. Equation (2.6) forces}_d₌_d

g0.

We define mq(G) as the number of cyclotomic q cosets of Ge =G_\{1_}. Theorem 2.2.5. Let e, µ give an n-adic code in Fq[G] then:

1. The orbits of the action of µ¯ on the cyclotomic cosets of Ge have length divisible by n.

2. The number of cyclotomic q-cosets of Ge is divisible by n. That is, mq(G) = 0

modulo n.

Proof. Clearly, part 2 is a consequence of part 1. Let e, µ give an n-adic code in

F_q[G]. Let Ω be an orbit of ¯µ on the cyclotomic cosets of Ge. Let _|µ_|=nl_∗_n

1 where

(n, n1) = 1.2 Note that µ0 = µn1 has order nl. Clearly, ¯(.) is an isomorphism of

2

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Aut(G), and _|µ¯0_| ₌_|_µ0_| ₌_nl_{. Since} _|_µ_¯₀_|₌ _nl_{, all orbits of} _µ0 _{on Ω have size a power} of n.

It suffices to show thatµ0 _{has no fixed points in Ω because then all orbits of}_µ0 _on Ω have size nl_{”, where} _l _≥ _{1; that is, their length is divisible by} _n_{. Hence, Ω must}

have size divisible by n.

Suppose µ0 has a fixed point in Ω, that is ¯µ0₍_Cg_{) =} _Cg_{. We will show that} _{e, µ}0 gives another n-adic code. Clearly, (n, n1) = 1 implies

{0,1, . . . , n₋1_} = _{0, n1,2∗n1, . . . ,(n−1)∗n1},

where the above equation is modulo n. Therefore,

{e, µ_·e, . . . , µn−1_·_e_} ₌ _{_{e, µ}n1 _·_{e, . . . , µ}n1∗(n−1)_·_e_}_. _(2.8)

By using an arithmetic argument, we can show thatnis the smallest power ofµ0 _such that (µ0₎n_·_e₌_e_{. Therefore,}

Stab_hµ0

i(e) = h(µ0)ni.

Equation (2.8) shows that

F_q[G] = _he, µ_·e, . . . , µn−1_·e_i

= _he, µ0_·e, . . . ,(µ0)n−1_·e_i, hdg0i = hei ∩ hµ·ei ∩ · · · ∩ hµ

n−1

·e_i

= _he_{i ∩ h}µ0_·e_{i ∩ · · · ∩ h}(µ0)n−1_·e_i.

Hence, e, µ0 _{gives an}_n_{-adic code.}

By corollary 2.2.4,Cg =C1 ={1}; however, this contradicts our choice of Ω which

is made of cyclotomic cosets of Ge and 1_∈/ Ge.

Theorem 2.2.6. There is an n-adic code in Fq[G]if and only if there is σ ∈Aut(G)

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Proof. Suppose there is ann-adic code inFq[G] given bye, µ. By the proof of theorem

2.2.5,σ= ¯µ0_{has order}_nl_{and the orbits of}_σ_{on the cyclotomic cosets of}_Ge_{have length}

divisible by n.

Conversely, let σ be such that the orbits of σ on the cyclotomic cosets of Ge have length divisible by n. Let Ω1, . . . ,ΩN be the orbits of σ on the cyclotomic cosets of

e

G. Let Ωi = {Cg_ji,₁, . . . , Cg_ji,N

i} where Ni is divisible by n and σ(Cgji,k) = Cgji,k+1

where k is taken modulo Ni. Let

Ωj,k = {Cj,k, Cj,k+n, . . . , Cj,k+l∗n, . . . , C_j,k₊₍Nj n−1)∗n}

where k +l_∗n is taken modulo Nj and k _{∈ {}1, . . . , n_}. Note that σ(Ωj,k) = Ωj,k+1

by construction, wherek is taken modulo n. Letdgi =

P

g∈C_gi eχg be the primitive idempotent corresponding to the cyclotomic

coset Cgi. Choose µ ∈ Aut(G) such that ¯µ = σ. Define dj,k = P

Cg∈Ωj,kdg. Since

the action of µ on the primitive idempotents is the same as the action of σ on the cyclotomic cosets, and σ(Ωj,k) = Ωj,k+1, we have µ· dj,k = dj,k+1. Define ek =

dg0 +d1,k +d2,k,+· · ·+dN,k, then µ·ek = ek+1 and all {e1, . . . , en} are distinct by construction.

Note that dj,k _∗dj0_,k0 = 0, unless j = j0 and k = k0. Also, dj,k ∗dg

0 = 0. Hence,

e1∗e2∗ · · · ∗en =dg0.

Also note thate1+· · ·+en−(n−1)e1∗· · ·∗en= 1. Thus,he1, . . . , eni=h1i=Fq[G],

and _he1i ∩ · · · ∩ heni=hdg0i; therefore, e1, µ gives ann-adic code. We observe that the constructed n-adic code e1, µ is also “reduced.”

As a corollary, we deduce that the “reduced” condition does not modify the solu-tion to the existence problem forn-adic codes.

Corollary 2.2.7. Fq[G] allows ann-adic code if and only ifFq[G]allows a “reduced” n-adic code.

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divisible by n. Applying theorem 2.2.6, once again, we can construct ann-adic code inFq[G] that is reduced.

(_⇐) This is clear.

Theorem 2.2.8. There is an n-adic code in Fq[G]if and only if there is σ ∈Aut(G)

such that _|σ_|=nl _and _σ _{fixes no cyclotomic coset of} _Ge_.

Proof. Suppose there is an n-adic code in Fq[G] given by e, µ. Then by the proof of

theorem 2.2.5, σ = ¯µ0 _{has order} _nl _and _σ _{fixes no cyclotomic coset of} _Ge_.

Conversely, suppose σ _∈ Aut(G) such that _|σ_| = nl _and _σ _{fixes no cyclotomic}

coset of Ge. Then, the orbits ofσ on the cyclotomic cosets of Ge have length divisible by n. Using the same construction of the second part of the proof of theorem 2.2.6, one can construct an n-adic code in Fq[G].

For the next sections,tk0(k) = |µk|inAut((Z/k0Z)), will denote the multiplicative

order of k modulo k0.

Corollary 2.2.9. Let G= (Z/sZ)_{× · · · ×}(Z/sZ) _'(Z/sZ)r_{. If there is an} _n_-adic

code in Fq[G], then mq(G) = s r

−1

ts(q) = 0 modulo n.

Proof. By theorem 2.2.5, it suffices to show thatmq(G) = s_tr−1

s(q). This will be done by

using the fundamental counting principle.

Define Ni =_|{g _∈Ge_|µqi(g) = µi_q(g) =g}|. Let t =ts(q), then

mq(G) = 1

t t−1

X

i=0

Ni.

Clearly, N0 =sr−1. Note thatNi = 0 fori∈ {1, . . . , t−1}; because, ifµqi(g) = g

then (qi ₋₁₎_∗_g _{= 0 in} _G_{. Thus} _qi ₋_{1 = 0 modulo} _|_g_| ₌ _s_{. This contradicts the}

choice of i. Thus, nog exists such that µqi(g) =g. Therefore,

mq(G) = 1

t{(s r

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Theorem 2.2.10. If there is an n-adic code in Fq[K] and in Fq[L] then there is an n-adic code in Fq[K×L].

Proof. By theorem 2.2.8, it suffices to show that there is γ _∈ Aut(K_×L) of order

|γ_|= nl_{, and} _γ _{fixes no cyclotomic coset of} _K^_×_L_{. By theorem 2.2.8, there are} _σ _∈ Aut(K) and τ _∈Aut(L) of orders_|σ_|=nl1 _and_|_τ_|₌_nl2_{, where}_σ_{fixes no cyclotomic} coset of Ke and τ fixes no cyclotomic coset of Le. Consider γ(k, l) = (σ(k), τ(l)). Clearly, _|γ_|=lcm(_|σ_|,_|τ_|) =lcm(nl1_{, n}l2_{) =}_nl_{. Let}_C

(k,l)denote the cyclotomic coset

ofK_×L containing (k, l). Now, supposeγ(C(k,l)) =C(k,l) whereC(k,l) is a cyclotomic

coset of K^_×L; then, (σ(k), τ(l)) = γ(k, l) = µi

q(k, l) = µqi(k, l) = (µ_qi(k), µ_qi(l)).

Thus, σ(k) = µqi(k) and τ(l) = µqi(l). Therefore, σ(Ck) = Ck and τ(Cl) = Cl. By

the choice ofσandτ,Ck=C1 andCl =C1. Thus,C(k,l) =C(1,1) and this contradicts

the choice of C(k,l) since C(1,1) is not a cyclotomic coset of K^×L. Therefore, γ fixes

no cyclotomic coset of K^_×L.

Theorem 2.2.11. Lete, µgive ann-adic code inFq[G]. SupposeK ⊂Gis a subgroup

such that µ(K) = K. Then, there is an n-adic code in Fq[K] and Fq[G/K].

Proof. By theorem 2.2.5, the orbits of σ = ¯µ on the cyclotomic cosets of Ge have length divisible by n. Consider the restriction of σ to K. Since µ(K) = K, it can be shown that ¯µ(K) = K. Thus, σ _∈ Aut(K). Since σ(K) = K, the orbit of a cyclotomic coset Ck ofK is made up of cyclotomic cosets of K. By assumption of σ, all orbits of σ on the cyclotomic cosets of Ke have length divisible byn. By theorem 2.2.6, there is an n-adic code in Fq[K].

To show the other result, consider the epimorphism ρ : G_→ G/K which can be extended to an algebra epimorphism ρ :Fq[G]→Fq[G/K]. As µ(K) =K, µ factors

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clearly an idempotent. Since ρ(µi_·_e_{) =} _φi_·_ρ₍_e_),

F_q[G/K] = _h1_i

= _hρ(e), ρ(µ_·e), . . . , ρ(µn−1_·e)_i = _he0, φ_·e0, . . . , φn−1 _·e0_i.

Also, note that:

e0_∗(φ_·e0)_{∗ · · · ∗}(φn−1_·e0) = ρ(e)_∗(ρ(µ_·e))_{∗ · · · ∗}(ρ(µn−1_·e)) = ρ(e_∗(µ_·e)_{∗ · · · ∗}(µn−1_·e)) = ρ(dg0)

= dg0.

Thus, _he0_{i ∩ h}φ_·e0_{i ∩ · · · ∩ h}φn−1_·_e0_i₌_h_dg 0i.

It suffices to show that the smallest power ofφthat fixese0 _is_n_{. Clearly,}_µn_·_e₌_e_,

sinceφn_·_e0 ₌_φn_·_ρ₍_e_{) =}_ρ₍_µn_·_e_{) =}_ρ₍_e_{) =}_e0_{. Suppose that there is 1}_≤_k _≤_n₋_{1 such} thatφk_·_e0 ₌_e0_{. Note that, since}_n _{is prime, (}_{k, n}_{) = 1; thus by choosing}_k0 _{such that}

k_∗k0 =n_∗n1+ 1, we have thate0 =φk∗k

0

·e0 =φn∗n1+1_·_e0 ₌_φ_·_e0_{. In particular}_h_e0_i₌

he0_{, φ}_·_e0_{, . . . , φ}n−1_·_e0_i ₌ _h₁_i ₌ _F

q[G/K]. By uniqueness of idempotents generating

ideals in Fq[G/K], we have that e0 = 1 in Fq[G/K]. Thus, ρ(e) = 1 in Fq[G/K]

which forces e to be an idempotent of Fq[K] alone. Note that since µ(K) = K, {e, µ_·e, . . . , µn−1_·_e_} _{are idempotents of} _F

q[K]. Thus Fq[G] = he, µ·e, . . . , µn−1·ei

is a subideal ofFq[K]. This is a contradiction; therefore,φ·e0 cannot equal e0. Thus,

no such k exists. Therefore,n is the smallest power of φ that fixes e0_.

Theorem 2.2.11 can be applied to any characteristic subgroup. In particular, it can be applied to the sylow subgroups of G, since these are characteristic as G is abelian. Thus, theorems 2.2.10 and 2.2.11 yield the following corollary.

Corollary 2.2.12. There is an n-adic code in Fq[G] if and only if there is an n-adic

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And also,

Corollary 2.2.13. Let s be a prime and let (s, q) = 1. If there is an n-adic code in

F_q[Z/sr_Z_]_{, then there is an} _n_{-adic code in} _F

q[Z/sZ].

Proof. LetG(1) =hsi ⊂(Z/srZ). Clearly, G(1) is a characteristic subgroup ofZ/srZ.

By applying theorem 2.2.10 toL= (Z/sr_Z₎_/G

(1) 'Z/sZ, the conclusion follows.

Corollary 2.2.12 reduces the existence problem to s-groups, where s is a prime. We will see that the existence problem in s-groups has different answers depending ons.

2.2.2

Existence in n-groups

For this section, we will let G= (Z/nZ)π1 _{× · · · ×}₍Z_/nrZ₎πr _{be an} _n_{-group. We will}

show that there are no n-adic codes in Fq[G].

Lemma 2.2.14. If G= (Z/nZ)π1_{, then there are no} _n_{-adic codes in} F

q[G].

Proof. By corollary 2.2.9, mq(G) = nπ1₋1

tn(q) = 0 modulo n. Note that modulo n we

have:

nπ1 ₋₁

tn(q) =

n₋1

tn(q){1 +n+· · ·+n

π1−1_}

= n−1

tn(q)

6

= 0.

This is a contradiction. Thus, there cannot be an n-adic code in F_q[G].

Theorem 2.2.15. Let G= (Z/nZ)π1 _{× · · · ×}₍Z_/nrZ₎πr _{be an abelian} _n_{-group, then}

there are no n-adic codes in F_q[G].

Proof. Define G(i) = {g ∈ G | |g| divides ni}. Clearly, G(i) is a characteristic

sub-group of G. Suppose there is an n-adic code in Fq[G]. Consider L = G/G(r−1) '

(Z/nZ)πr_{. By theorem 2.2.11, there is an} _n_{-adic code in} F

q[L]. This contradicts

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2.2.3

Existence in s-groups with

(

s

,

n

) =

1

The existence problem depends on the value of s modulo n and on the parity of s

modulo 2. The parity of s modulo 2 is important, since (Z/sr_Z₎∗ _{is not cyclic when}

s = 2, whereas it is when s is an odd prime. Because of this, the proofs have to be modified. We note that by the previous section, we can assume that s₆=n.

2.2.3.1 Existence in s-groups, (s,2) =1 First, we will consider G=Z/sZ. Let t =ts(q).

Theorem 2.2.16. There is ann-adic code inF_q[Z/sZ]if and only if s−1

t =mq(Z/sZ) =

0 modulo n.

Proof. Suppose there is an n-adic code in Fq[Z/sZ], then by theorem 2.2.5 we have mq(Z/sZ) = 0 modulo n. Note that by corollary 2.2.9 we have s−_t1 =mq(Z/sZ).

Conversely, suppose N = s−_t1 = n_∗n1 is divisible by n. Let {Cg1, . . . , CgN} be

all the cyclotomic cosets of Ge. By theorem 2.2.6, it suffices to find σ _∈Aut(G) such that the orbits of σ on the cyclotomic cosets of Ge have length divisible by n. It will be shown that σ =µl, whereAut(Z/sZ) = (Z/sZ)∗ ₌_h_µl_i _{will do the job.}

Consider, H = hµli

hµqi =

(Z/sZ)∗

hqi = hli

hqi. Note that every coset of H is a cyclotomic coset. This can be seen by considering,3

Cg = {[g],[g∗q], . . . ,[g∗qt−1]}

= µg({[1],[q], . . . ,[qt−1]})

= µg(hµqi).

Since H is cyclic and generated by µl, this shows that µl acts transitively on the cosets of H. Thus, µl acts transitively on the cyclotomic cosets ofGe. Therefore, the action of µl on the cyclotomic cosets of Ge is one orbit of length N. Since N = 0 modulo n, the conclusion follows.

Now, we consider G=Z/sr_Z_.

3

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Theorem 2.2.17. There is an n-adic code in Fq[Z/srZ] if and only if there is an n-adic code in Fq[Z/sZ].

Proof. Suppose there is an n-adic code in Fq[Z/srZ], then by corollary 2.2.13 there

is an n-adic code in Fq[Z/sZ].

Conversely, suppose there is an n-adic code in Fq[Z/sZ]. Let G = Z/srZ. Since s ₆= 2, Aut(G) = _hµl_i is cyclic. Following the same construction as in the proof of theorem 2.2.16, we will show that the action of µl on the cyclotomic cosets of Ge has orbits of length divisible by n.

Define Si = _{g _∈ Ge _| g = sr−i _∗_g

1,(g1, s) = 1}. Clearly, Si is a characteristic

subset ofG. Thus,Si is the union of cyclotomic cosets of Ge. Also, Ge=S1∪ · · · ∪Sr.

Define Ni to be the number of cyclotomic cosets of Si. First, we will calculate

Ni = si−_t1(s−1)

si(q) . Second, we will show thatµl acts transitively on the cyclotomic cosets

of Si and that Ni is divisible by n if and only if N1 is divisible by n. By theorem

2.2.16,N1 = _ts_s−₍_q1₎ is divisible by n. Hence, this will show that the action of µl on the

cyclotomic cosets Ge is made up of r orbits, each orbit of lengthNi where all Nis are divisible by n. By theorem 2.2.6, the conclusion will follow.

Let Aut(Z_si) = (Z/siZ)∗ be the units of the ring Z/siZ. Define Hi = (Z/s i_Z)∗

hµqi and Ki = (Z/si_Z₎∗_{. Consider} _Cg _{a cyclotomic coset of} _Si_{. Then,} _g ₌ _sr−i _∗_g

1, where

(g1, s) = 1. Note that we think of g1 as a unit in Z/siZ. Consider,

Cg = _{[sr−i_∗g1],[sr−i∗g1∗q], . . . ,[sr−i∗g1∗qtsi(q)−1]}

= µsr−i(µg

1({[1],[q], . . . ,[q

t_si(q)−1_]

})) = µsr−i(µg

1hµqi).

The above implies that Cg is the image under µsr−i of a coset of Hi. This shows

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calculated directly as:

|Hi_| = |(Z/s

i_Z₎∗_|

|hµq_i|

= s

i−1₍_s₋₁₎

tsi(q) .

Since µl generates (Z/sr_Z₎∗_{, it also generates (}_Z_/si_Z₎∗ _{for every 1} _≤ _i _≤ _r_.4 _Thus,

µl acts transitively on the cosets of Hi and therefore on the cyclotomic cosets ofSi.

Now we show that _|Hi_|is divisible by n if and only if _|H1|is divisible by n. This

will show that Ni is divisible by n if and only if N1 is divisible by n, and the proof

will be finished.

Lemma 2.2.18. q is an n-adic residue modulo s if and only if q is an n-adic residue modulo si _{for every} _i_.5

Proof. This is a result from Number Theory and can be shown in the following way. Clearly, if q is an n-adic residue modulo si_{, then it is an} _n_{-adic residue modulo} _s_.

Conversely, suppose q is an n-adic residue modulo s. Consider the equation:

Xn₋q = 0, (2.9)

in the s-adic numbers Qs, where s is a prime. Since q is an n-adic residue modulo s, Equation (2.9) has a solution in the residue field Zs/(sZs) =Fs of Qs.6 Note that f(x) = xn₋_q _{has no double roots in} _F

s as f0(x) = n∗xn−1 and (f(x), f0(x)) = 1

in Fs[x]. Thus, f(x) = (x − xo)g(x), where xo ∈ Fs; and g(x) ∈ Fs[x], where

(g(x), x₋xo) = 1 in Fs[x]. By Hensel’s lemma, this factorization can be lifted to

a factorization in Zs[x]. That is, there is Xo ∈ Zs and h(x) ∈ Zs[x] such that: f(x) = (x₋X0)h(x),x−Xo =x−xo modsZs[x]; where, h(x) = g(x) mod sZs[x].

Thus,Xn ₌_q_{has a solution in the}_s_{-adic integers, namely}_Xo_{. Through the canonical}

projection from Zs → Zs/(siZs) = Z/siZ, we find a solution of Xn = q in Z/siZ.

4

µlwhen viewed as an element of (Z/siZ)∗is basicallyµlowherel=lomodulosi(1≤lo≤si−1).

5

qis ann-adic residue modulosi _{if and only if there is}_x

∈Z/si_Z_{such that}_xn₌_q_mod_si_.

6

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Thus, q is an n-adic residue modulo si_.

Let Ki = (Z/si_Z₎∗_{. Suppose that} _|_Hi_| ₌ si−1₍_s₋₁₎

t_si(q) is divisible by n. Note that,

since (s, n) = 1 and n divides _|Hi_| = si−_t1(s−1)

si(q) , we have that n divides s−1

t_si(q). In

particular, n divides s₋1; therefore, n divides _|Kj_|=sj−1₍_s₋_{1) for all} _j_.

Since Kj is cyclic andn divides _|Kj_|, we have that Kj Kn

j =

Z/nZ, where Kn

j ={g ∈ Kj _|xn ₌_g,_{for some} _x_∈_Kj_}_.7 _{Therefore, we have the following lemma.}

Lemma 2.2.19. If n divides _|Hi_| for some i, then n divides _|Kj_| for all j and Ki Kn

i = Z/nZ.

Now we are ready to prove our final claim.

Lemma 2.2.20. The order of Hi is divisible by n if and only if the order of H1 is

divisible by n.

Proof. Suppose_|Hi_|is divisible byn. By lemma 2.2.19,ndivides_|Kj_|and Kj Kn

j = Z/nZ

for all j.

Since Ki is cyclic, we can apply the following claim toKi that we include without proof.

Claim 2.2.21. Suppose H, K are subgroups of a cyclic group G. If _|H_| divides _|K_|

then H _⊂K.

Consider,

|hµq_{i| ∗ |}Hi_| = _|Ki_|

=

¯ ¯ ¯ ¯_KKin

i

¯ ¯ ¯

¯∗ |Kin|

= n_{∗ |}K_in_|.

i |. By claim 2.2.21, we have that hµq_{i ⊂}Kn

i . In particular, q is an n-adic residue modulo si.

7

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By lemma 2.2.18, q is an n-adic residue modulo s. Thus µq _∈Kn

1. Consider:

|H1| =

¯ ¯ ¯ ¯

K1

hµq_i

¯ ¯ ¯ ¯ = ¯ ¯ ¯ ¯_KK1n

1 ¯ ¯ ¯ ¯∗ ¯ ¯ ¯ ¯K n 1

hµq_i

¯ ¯ ¯ ¯

= n_∗

¯ ¯ ¯ ¯ K n 1

hµq_i

¯ ¯ ¯ ¯.

Thus, n divides _|H1|.

Conversely, suppose n divides _|H1|. By lemma 2.2.19, we have that n divides Kj

and Kj Kn

j = (

Z/nZ) for all j. By considering,

|hµq_{i| ∗ |}H1| = |K1|

=

¯ ¯ ¯ ¯_KK1n

1

¯ ¯ ¯

¯∗ |K1n|

= n_{∗ |}K₁n_|,

one can conclude that _|hµ_qi| divides _|Kn

1| since n divides |H1|. Thus, q is an n-adic

residue modulosby claim 2.2.21 about cyclic groups. By lemma 2.2.18,qis ann-adic residue modulosi_{. Using the same argument as above, by considering the equation:}

|Hi_| =

¯ ¯ ¯ ¯_hKi_µq_i

¯ ¯ ¯ ¯ = ¯ ¯ ¯ ¯_KKin

i ¯ ¯ ¯ ¯∗ ¯ ¯ ¯ ¯K n i hµ_qi

¯ ¯ ¯ ¯

= n_∗

¯ ¯ ¯ ¯ K n i hµq_i

¯ ¯ ¯ ¯,

we conclude that n divides Hi.

Now we consider general s-groups. Let G = (Z/sZ)π1 _{× · · · ×}₍Z_/srZ₎πr_{. Note}

that if mq(Z/sZ) = s−1

ts(q) = 0 modulo n, then by theorem 2.2.16 there is an n-adic

code in Fq[Z/sZ]. By theorem 2.2.17, there is an n-adic code in Fq[Z/srZ], for all r.

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Corollary 2.2.22. If mq(Z/sZ) = 0 modulo n and G= (Z/sZ)π1 _{× · · · ×}₍Z_/srZ₎πr

is an s-group with (s, n) = 1, s odd, then there is an n-adic code in Fq[G].

However, what happens whenmq(Z/sZ)₆= 0 modulon? This is answered partially by the following theorem.

Theorem 2.2.23. Suppose mq(Z/sZ)= 0₆ modulon, (s, n) = 1, ands an odd prime. If there is an n-adic code in G= (Z/sZ)π1 _{× · · · ×}₍Z_/srZ₎πr_{, then either:}

1. If s = 1 modulo n, then πk = 0 modulo n for all k, 2. If s ₆= 1 modulo n, then tn(s) divides πk for all k.

Proof. Consider the case G = (Z/sZ)π1_{. Suppose there is an} _n_{-adic code in} F

q[G].

By theorem 2.2.5, sπ₁

−1

ts(q) = mq(G) = 0 modulo n. Thus, s−1

ts(q){1 +s+· · ·+s

π1−1_}₌

sπ₁₋₁

ts(q) = 0 modulo n. Since s−1

ts(q) 6= 0 modulo n and n is a prime, we can divide by s−1

ts(q) in the previous equation to get 1 +s+· · ·+s