A New Approach to the Kasami Codes of Type 2

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A New Approach to the Kasami Codes of Type 2

Minjia Shi, Denis Krotov, Patrick Sole

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Minjia Shi, Denis Krotov, Patrick Sole. A New Approach to the Kasami Codes of Type 2. 2019.

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IEEE TRANSACTIONS ON INFORMATION THEORY 1

A New Approach to the Kasami Codes of Type 2

Minjia Shi , Denis S. Krotov , Member, IEEE, and Patrick Solé

Abstract— The dual of the Kasami code of lengthq²−1, with

1

q a power of 2, is constructed by concatenating a cyclic MDS

2

code of length q + 1 over Fq with a Simplex code of length

3

q −1. This yields a new derivation of the weight distribution

4

of the Kasami code, a new description of its coset graph, and

5

a new proof that the Kasami code is completely regular. The

6

automorphism groups of the Kasami code and the relatedq-ary

7

MDS code are determined. New cyclic completely regular codes

8

over finite fields a power of 2, generalized Kasami codes, are

9

constructed; they have coset graphs isomorphic to that of the

10

Kasami codes. Another wide class of completely regular codes,

11

including additive codes, as well as unrestricted codes, is obtained

12

by combining cosets of the Kasami or generalized Kasami code.

13

Index Terms— Kasami codes, completely regular codes, cyclic

14

codes, concatenation, automorphism group.

15

I. INTRODUCTION

16

D

17

studied class of structured graphs due to their many con-

18

nections with codes, designs, groups and orthogonal polyno-

19

mials [1], [6]. Since the times of Delsarte [7], a powerful way

20

to create distance-regular graphs, especially in low diameters,

21

has been to use the coset graph of completely regular codes.

22

A linear or additive code is completely regular if the weight

23

distribution of each coset solely depends on the weight of its

24

coset leader. Many such examples from Golay codes, Kasami

25

codes, and others, can be found in [6]. A recent survey is [5].

26

In this note we focus on the Kasami code of type (ii) in the

27

sense of [6, § 11.2]. This code was also studied in [3]. It is a

28

cyclic code of length 2^2m−1 with the two zeros α,α^1+2m,

29

where α denotes a primitive root of F₂^2m. We call such a

30

code a classical Kasami code. We give a new construction

31

by concatenating a cyclic MDS code with a binary Simplex

32

code. This gives a new derivation of its weight distribution,

33

a non trivial calculation in [13], and a characterization of its

34

automorphism group.

35

AQ:1 Manuscript received December 28, 2018; revised August 2, 2019; accepted October 14, 2019. This work was supported in part by the National Natural Science Foundation of China under Grant 61672036, in part by the Excellent Youth Foundation of Natural Science Foundation of Anhui Province under Grant 1808085J20, in part by the Academic Fund for Outstanding Talents in Universities under Grant gxbjZD03, and in part by the Program of Fundamental Scientific Researches of the Siberian Branch of the Russian Academy of Sciences under Project 0314-2019-0016.

M. Shi is with the Key Laboratory of Intelligent Computing and Signal Processing of Ministry of Education, School of Mathematical Sciences, Anhui University, Hefei 230601, China (e-mail: smjwcl.good@163.com).

D. S. Krotov is with the Sobolev Institute of Mathematics, 630090 Novosi- birsk, Russia (e-mail: krotov@math.nsc.ru).

AQ:2 P. Solé is with I2M, Centrale Marseille, Aix Marseille University, CNRS, AQ:3 13397 Marseille, France (e-mail: sole@enst.fr).

Communicated by V. Sidorenko, Associate Editor for Coding Theory.

Digital Object Identifier 10.1109/TIT.2019.2949609

More generally, we construct generalized Kasami codes ³⁶ of lengths ^q_p−1²⁻¹, when q is an even power of p, and p ³⁷ itself a power of 2. This is obtained by concatenation of ³⁸ a q-ary cyclic MDS code with a p-ary Simplex code. The ³⁹ resulting code is cyclic with two zerosα,α^(p−1)(1+q), where ⁴⁰ α denotes a primitive root of F_q². When p > 2, we obtain ⁴¹ in that way infinitely many new cyclic completely regular ⁴² codes with a coset graph isomorphic to the coset graph of a ⁴³

Kasami code. ⁴⁴

We emphasize that one of the main contributions of our ⁴⁵ research is in applying the concatenation with the Simplex ⁴⁶ code to cyclic codes, with the result being cyclic too. In ⁴⁷ general, the concatenation approach in the construction of 48

linear completely regular codes is known, see [16], [17], [4], 49

[5]. In particular, in [4], the concatenation of dual-distance-3 50

MDS codes with the Simplex code was considered (note that a ⁵¹ distance-3 MDS code, even non-linear, is always completely ⁵² regular [12, Cor. 6]); the result is a two-weight q-ary code, ⁵³ whose dual is a covering-radius-2 completely regular code. ⁵⁴ We use a similar approach, but start with a cyclic dual- ⁵⁵ distance-4 MDS code, and prove also the cyclicity of the ⁵⁶ resultingp-ary (pis a power of two) completely regular code ⁵⁷ of covering radius 3, which is known as the Kasami code ⁵⁸ if p = 2, and is a new cyclic completely regular code if ⁵⁹ p > 2. In fact, concatenation with the Simplex code can be ⁶⁰ applied as well to produce a completely regular p-ary code ⁶¹ from a linear completely regular q-ary code, q = p^k, with ⁶² the same intersection array, independently on the concrete ⁶³ parameters. Moreover, even more general approach guarantees ⁶⁴ that from any unrestricted (not necessarily linear or additive)q- ⁶⁵ ary completely regular code (and even an equitable partition) ⁶⁶ we can construct a completely regular code with the same ⁶⁷ intersection matrix over any alphabetpsuch thatqis a power ⁶⁸ of p (the alphabet sizes p and q are not required to be ⁶⁹ prime powers). This approach is based on the existence of ⁷⁰ a locally bijective homomorphism, known as a covering, see, ⁷¹ e.g., [8, Sect. 6.8], between the Hamming graphsH(n, q)(in ⁷² the role of the cover) and H(_p−1^q−1n, p) (in the role of the ⁷³ target) of the same degree. With the inverse homomorphism, ⁷⁴ an equitable partition or completely regular code of the target ⁷⁵ graph automatically maps to an equitable partition or com- ⁷⁶ pletely regular code, respectively, of the cover graph, with 77

the same intersection matrix. This combinatorial approach is 78

very powerful and allows to connect equitable partitions of 79

different graphs, especially translation graphs (Cayley graphs ⁸⁰ over abelian groups), such as, for example, bilinear forms ⁸¹ graphs and Hamming graphs. However, it does not always ⁸² allow to track some nice algebraic properties of the codes ⁸³

such as cyclicity. ⁸⁴

This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see http://creativecommons.org/licenses/by/4.0/

The material is arranged as follows. The next section

85

collects the notions and notations needed for the rest of the

86

paper. Section III describes the concatenation approach and

87

the inner code used. Section IV gives a new description of the

88

classical Kasami codes, including their weight distributions,

89

and their automorphism groups. Section V describes the (new)

90

generalized Kasami codes obtained by allowing a more general

91

alphabet in the outer code. This includes a determination of

92

the weight distribution, and of the coset graphs. Section VI

93

provides a combinatorial way to derive new completely regular

94

codes from old, and applies it to the generalized Kasami codes.

95

Section VII recapitulates the path followed, and points out new

96

directions.

97

II. DEFINITIONS ANDNOTATION

98

A. Graphs

99

All graphs in this note are finite, undirected, connected,

100

without loops or multiple edges. In a graph Γ, the neighbor-

101

hoodΓ(¯v)of the vertexv¯is the set of vertices connected tov.¯

102

Thedegreeof a vertex¯vis the size ofΓ(¯v). A graph isregular

103

if every vertex has the same degree. Thei-neighborhoodΓ_i(¯v)

104

is the set of vertices at geodetic distance i tov. A graph is¯

105

distance regular(DR) if for every pair or vertices u¯andv¯at

106

distance iapart the quantities

107

b_i=|Γi+1(¯u)∩Γ(¯v)|, c_i=|Γi−1(¯u)∩Γ(¯v)|,

108

which are referred to as the intersection numbers of the

109

graph, solely depend on i and not on the special choice of

110

the pair (¯u,v). The¯ automorphism group of a graph is the

111

set of permutations of the vertices that preserve adjacency.

112

The Hamming graph H(n, q) (we consider only the case

113

of prime power q) is a distance-regular graph on Fⁿ_q, two

114

vectors being connected if they are at Hamming distance

115

one. It is well known [14] that any automorphism of the

116

Hamming graph acts as a permutation π of coordinates and,

117

for every coordinate, a permutation σ_i of Fq; i.e., σ◦ π:

118

(v₁, . . . , v_n)→(σ₁(v_π−1(1)), . . . , σ_n(v_π−1(n))).

119

B. Equitable Partitions, Completely Regular Codes

120

A partition (P₀, . . . , P_k) of the vertex set of a graph Γ is

121

called anequitable partition(also known as regular partition,

122

partition design, perfect coloring) if there are constants S_ij

123

such that

124

|Γ(¯v)∩P_j|=S_ij for everyv¯∈P_i, i, j= 0, . . . , k.

125

The numbers S_ij are referred to as the intersection

126

numbers, and the matrix (S_ij)^k_i,j=0 as the intersec-

127

tion (or quotient) matrix. If the intersection matrix

128

is tridiagonal, then P₀ is called a completely regu-

129

lar code of covering radius k and intersection array

130

(S₀₁, S₁₂, . . . , S_k−1k;S₁₀, S₂₁, . . . , S_{k k−1}). That is to say,

131

a set of verticesCis a completely regular code if the distance

132

partition with respect to C is equitable. As was proven by

133

Neumaier [15], for a distance regular graph, this definition

134

is equivalent to the original Delsarte definition [7]: a code

135

C is completely regular if the outer distance distribution

136

(|C∩Γ_i(¯v)|)i=0,1,2,... ofCwith respect to a vertexv¯depends

137

only on the distance fromv¯toC.

138

C. Linear and Additive Codes ¹³⁹

A linear code (that is, a linear subspace ofFⁿ_q) of lengthn, ¹⁴⁰ dimensionk, minimum distanced over the fieldFq is called ¹⁴¹ an [n, k, d]_q code. An[n, k, d]_q code is called an MDS code ¹⁴² if d = n−k+ 1. The duality is understood with respect ¹⁴³ to the standard inner product. Let A_w denote the number of ¹⁴⁴

codewords of weightw. ¹⁴⁵

Linear codes are partial cases of the additive codes, which ¹⁴⁶ are, by definition, the subgroups of the additive group ofFⁿ_q. A ¹⁴⁷ cosetof an additive codeCis any translate ofCby a constant ¹⁴⁸ vector. A coset leader is any coset element that minimizes ¹⁴⁹ the weight. Theweight of a coset is the weight of any of its ¹⁵⁰ leaders. Thecoset graphΓ_C of an additive codeC is defined ¹⁵¹ on the cosets of C, two cosets being connected if they differ ¹⁵²

by a coset of weight one. ¹⁵³

Lemma 1 (see, e.g., [6]). An additive code with dis- ¹⁵⁴ tance at least 3 is completely regular with intersec- ¹⁵⁵ tion array {b0, . . . , b_ρ−1;c₁, . . . , c_ρ} if and only if the ¹⁵⁶ coset graph is distance-regular with intersection numbers ¹⁵⁷ b₀, . . . , b_ρ−1, c₁, . . . , c_ρ. ¹⁵⁸ The weight distribution of a code is displayed as 159

[0,1,· · ·,w, A_w,· · ·], where A_w denotes the number of 160

codewords of weightw. ¹⁶¹

D. Cyclic Codes ¹⁶²

A cyclic codeC of lengthn overFq is, up to polynomial ¹⁶³ representation of vectors, an ideal of the ringFq[x]/(xⁿ−1). ¹⁶⁴ All ideals are principal with a unique monic generator, called ¹⁶⁵ the generator polynomialofC, and denoted by g(x). If this ¹⁶⁶ polynomial hastirreducible factors overFq, then the codeC ¹⁶⁷ is called a cyclic codewithtzeros. In that case, the dualC^{⊥ 168} admits a trace representation witht terms of the form ¹⁶⁹

c(a₁, . . . , a_t) = ^t

i=1

Tr(a_iα^j_i) _n

j=1, ¹⁷⁰

where ¹⁷¹

• k_i is the degree of the factor indexediofg(x), ¹⁷²

• Tr = Tr_qki/q is the trace fromF_qki down toFq, ¹⁷³

• a_i is arbitrary inF_qki, ¹⁷⁴

• α_i is a primitive root inF^×_qki, ¹⁷⁵ In general the relative trace from Fq^r to Fq, where q is a ¹⁷⁶

power of a prime, is defined as ¹⁷⁷

Tr_qr/q(z) =z+z^q+· · ·+z^qr−1. ¹⁷⁸ If q is prime, then Tr_qr/q(z) is the absolute trace. See the ¹⁷⁹ theory of the Mattson–Solomon polynomial in [13] for details ¹⁸⁰

and a proof. ¹⁸¹

E. Automorphisms ¹⁸²

The automorphism group Aut(C) of a q-ary code C of ¹⁸³ lengthnis the stabilizer ofC in the automorphism group of ¹⁸⁴ the Hamming graph H(n, q). Note that every automorphism ¹⁸⁵ acts as the composition of a coordinate permutation and a ¹⁸⁶ permutation of the alphabet in every coordinate. ¹⁸⁷

SHIet al.: NEW APPROACH TO THE KASAMI CODES OF TYPE 2 3

We will say that a subgroupA of the automorphism group

188

of the Hamming graph H(n, q) acts imprimitively on the

189

coordinates if the set of coordinates can be partitioned into

190

subsets, called blocks, and denoted by N₁, …, N_t, 1 <

191

t < n, such that the coordinate-permutation part of every

192

automorphism from A sends every block to a block. If there

193

is no such a partition, then the action of the group on the

194

coordinates is primitive (for example, this is the case for

195

Aut(C)of any cyclic code C of prime length).

196

We denote byAut0(C)the stabilizer of the all-zero word in

197

Aut(C). Themonomial automorphism group AutM(C) of a

198

q-ary codeC is defined as the set of allmonomial transforms

199

that leave the code wholly invariant [11]. A monomial trans-

200

form acting onFⁿ_q is the product of a permutation matrix by

201

an invertible diagonal matrix, both in F^n×n_q . The subgroup

202

consisting of all such transforms with trivial diagonal part

203

is called the permutation part of the automorphism group.

204

The groupAutS(C)consists of thesemilinear automorphisms

205

of a q-ary code C, where each semilinear automorphism is

206

the composition τ ◦μ of a monomial transform μ and an

207

automorphismτ of the fieldFq (acting simultaneously on the

208

values of all coordinates) such that τ◦μ(C) =C. Note that

209

for any linear codeC, the groupsAutM(C)andAutM(C^⊥)

210

are always isomorphic, as well as the groups AutS(C) and

211

AutS(C^⊥); the same is not necessarily true forAut orAut0.

212

If q is not prime and Fq has a subfield Fp, we also use

213

the notation AutL_p(C) to denote the subgroup of Aut0(C)

214

consisting of only transformations that are linear overFp. Any

215

automorphism from AutL_p(C) acts as the composition of

216

a coordinate permutation and a Fp-linear permutation (from

217

GL(log_pq, p)) of Fq in every coordinate. In particular, if q

218

is a power of 2, then we have AutM(C) ≤ AutS(C) ≤

219

AutL₂(C) ≤ Aut0(C) ≤ Aut(C), and if q = 2, then the

220

first four of these groups are the same (the last group, for an

221

additive code, is the product of Aut0(C) and the group of

222

translations by codewords).

223

III. CONCATENATIONWITH THESIMPLEXCODE

224

Concatenation is the replacement of the symbols of the

225

codewords of some code, called the outer code, by the

226

codewords of some other code, called theinner code. The role

227

of the inner code in our study is played by a Simplex code;

228

moreover, we mainly focused on the case when this code is

229

cyclic.

230

Let p be a prime power; let q = p^k. Let ξ be a root of

231

unity of order ^q−1_p−1 in Fq. We assume that k andp−1 are

232

coprime; hence, so are ^q−1_p−1 andp−1 (indeed, by induction,

233

p^k−1

p−1 ≡kmodp−1). It follows that

234

F^×_q =

bξ^l|b∈F^×_p, l∈ {1, . . . ,^q−1_p−1}

(1)

235

and, in particular, all powers ofξare mutually linear indepen-

236

dent overFp. Then, the matrix(ξ¹, . . . , ξ^q−1p−1), considered as

237

ak×^q−1_p−1 matrix overFp, is a generator matrix of a Simplex

238

code of sizeq, where the distance between any two different

239

codewords is p^k−1 [11, Th. 2.7.5].

240

Encoding the Simplex code. Let L be a nondegenerate ²⁴¹ Fp-linear function fromFq toFp. Any such function can be ²⁴² represented asL(·)≡Tr_q/p(a·) for somea∈Fq, and in our 243

context we can always assume that L is Tr_q/p. Define φ : 244

Fq →Fp^q−1p−1 by ²⁴⁵

φ(z) =

L(zξ¹), . . . , L(zξ^q−1p−1)

; ²⁴⁶

expand its action toFⁿ_q coordinatewise, and to any set of words ²⁴⁷

inFⁿ_q wordwise. ²⁴⁸

Lemma 2. For any two words y,¯ z¯ in Fⁿ_q, we have ²⁴⁹ d(φ(¯y), φ(¯z)) =p^k−1d(¯y,z). I.e.,¯ φis a scaled isometry. ²⁵⁰ Proof: Immediately from the fact that the Simplex code ²⁵¹ is a one-weight code with nonzero weightp^k−1, we have ²⁵² d(φ(y₁, . . . , y_n), φ(z₁, . . . , z_n)) ²⁵³

=d

(φ(y₁), . . . , φ(y_n)),(φ(z₁), . . . , φ(z_n))

254

= n i=1

d(φ(y_i), φ(y_i)) = n i=1

p^k−1d(y_i, y_i) ²⁵⁵

=p^k−1d((y₁, . . . , y_n),(z₁, . . . , z_n)). ²⁵⁶

257

Lemma 3. Assume that M = (M_ij)^m_i=1ⁿ_j=1 is an m×n ²⁵⁸ generator matrix of a codeC inFⁿ_q. Then the matrix ²⁵⁹

Φ(M) =

⎛

⎜⎝

M₁₁ξ¹ ... M₁₁ξ^q−1p−1 M₁₂ξ¹ . . . M_1nξ^p−1q−1 ... ... ... ... . . . ...

M_m1ξ¹ ... M_m1ξ^q−1p−1 M_m2ξ¹ . . . M_mnξ^q−1p−1

⎞

⎟⎠, ²⁶⁰

considered as a km× ^q−1_p−1n matrix over Fp, is a generator ²⁶¹ matrix of the codeφ(C) inFp^p−1q−1n. ²⁶² Proof: The proof is straightforward. Any codeword ¯c of ²⁶³ the code generated by Φ(M) is a linear combination of its ²⁶⁴

rows, and so is of the form ²⁶⁵

¯ c=

m i=1

L_i(M_i1ξ¹), . . . , L_i(M_inξ^q−1p−1)

266

for some Fp-linear functions L_i : Fq → Fp. SinceL_i(·) ≡ ²⁶⁷ L(a_i·)for somea_i∈Fq, we have ²⁶⁸

¯ c =

m i=1

L(a_iM_i1ξ¹), . . . , L(a_iM_inξ^p−1q−1)

269

= m i=1

φ(a_iM_i1), . . . , φ(a_iM_in)

270

= m i=1

φ(a_iM_i1, . . . , a_iM_in)

271

= φ ^m

i=1

a_i(M_i1, . . . , M_in)

. ²⁷²

That is,c¯is aφ-image of some codeword of C. ²⁷³ The following lemma provides an alternative explanation ²⁷⁴ to the fact that completely regular codes with identical coset ²⁷⁵ graphs can exist over different alphabets [17]. ²⁷⁶

Lemma 4. The coset graph Γ_C⊥ of the dualC^⊥ of a linear

277

codeC inFⁿ_q and the coset graphΓ_φ(C)⊥ of the dualφ(C)^⊥

278

of its imageφ(C) inFp^p−1q−1n are isomorphic.

279

Proof: We first observe that the number of cosets ofC^⊥

280

coincides with size of the codeC, which is the same as the size

281

of the codeφ(C), which is, in its turn, the number of cosets of

282

φ(C)^⊥. That is, the considered graphs have the same number

283

of vertices.

284

LetCbe a code overFq with a generator matrixM, which

285

is a check matrix of the dual code C^⊥. The cosets of C^⊥

286

naturally correspond to the syndromes under the action of

287

the check matrix M. By the definition of the coset graph

288

Γ_C⊥, two cosets are adjacent in the coset graph if and only

289

if the difference between the corresponding syndromes is the

290

syndrome of a weight-1 word. We call the syndromes of the

291

weight-1 words the connecting syndromes because they play

292

a connecting role for the coset graph (this terminology is

293

in agree with that for a more general class of graphs called

294

Cayley graphs). For the check matrixM = (M_ij)^m_i=1ⁿ_j=1 over

295

Fq, the connecting syndromes are

296

a(M_1j, . . . , M_mj)^T, j∈ {1, . . . , n}, a∈F^×_q (2)

297

(corresponding to the weight-1 words with a in the j-th

298

position).

299

Now, consider the matrixΦ(M), which, by Lemma 3, is a

300

generator matrix forφ(C)and a check matrix forφ(C)^⊥. The

301

connecting syndromes are

302

b(ξ^lM_1j, . . . , ξ^lM_mj)^T, (3)

303

j∈ {1, . . . , n}, l∈

1, . . . ,q−1 p−1

, b∈F^×_p,

304

corresponding to the weight-1 words with b in the position

305

(j −1)_p−1^q−1 +l. Since {bξ^l | b ∈ F^×_p, l ∈ {1, . . . ,^q−1_p−1}}

306

coincides withF^×_q (see (1)), we find that the sets of connecting

307

syndromes (2), (3) for both matrices coincide; call this setS.

308

Two cosets of C^⊥ are adjacent in Γ_C⊥ if and only if

309

the corresponding syndromes have the difference in S. Two

310

cosets of φ(C)^⊥ are adjacent in Γ_φ(C)⊥ if and only if

311

the corresponding syndromes have the difference in S. So,

312

indexing the cosets by the syndromes induces an isomorphism

313

between the Γ_C⊥ andΓ_φ(C)⊥.

314

Corollary 1. If the code C^⊥ is completely regular, then the

315

code φ(C)^⊥ is completely regular with the same intersection

316

array.

317

In the rest of this section, we establish important relations

318

between the automorphism groups of a code and its concate-

319

nated image.

320

A. Automorphism Group

321

Lemma 5. The groupAutL_p(C)is isomorphic to a subgroup

322

ofAutM(φ(C)) such that the action of this subgroup on the

323

coordinates is imprimitive with the blocks

324

1, . . . ,q−1 p−1

, . . . ,

q−1

p−1(n−1)+1, . . . ,q−1 p−1n

.

325

Proof: The action of the monomial automorphism group ³²⁶ on a codewordcofC can be represented as a permutation of ³²⁷ the coordinates and, for every ifrom 1 tonthe action of an 328

element of GL(m, p) on of the value in the i-th coordinate, 329

understood as an element ofF^m_p . We first consider these two 330

actions separately. ³³¹

(i) A permutation of coordinates corresponds to a permuta- ³³²

tion of blocks forφ(C): we see that ³³³

φ(z_π−1(1), . . . , z_π−n(n)) = (φ(z_π−1(1)), . . . , φ(z_π−n(n))) ³³⁴ is obtained fromφ(z₁, . . . , z_n)by sending each i-th block of ³³⁵ coordinates to the place of theφ(i)-th block. ³³⁶ (ii) Next, we show that the action of aFp-linear permutation ³³⁷ on the value z_i in the i-th coordinate corresponds to a ³³⁸ monomial transform inside the i-th block of φ(z₁, . . . , z_n). ³³⁹ Here, a crucial observation if the following well-known fact. ³⁴⁰ (*) Any bijective linear transform of the Simplex code ³⁴¹ is equivalent to the action of some of its monomial auto- ³⁴² morphisms. This fact is straightforward from the form of a ³⁴³ generator matrix of the Simplex code, which consists of a ³⁴⁴ complete collection of pair-wise linearly independent columns ³⁴⁵ of heightm. Applying a bijective linear transform to the rows, ³⁴⁶ we obtain another generator matrix with the same property. It ³⁴⁷ is obvious that one such matrix can be obtained from another ³⁴⁸ one by permuting columns and multiplying each column by a ³⁴⁹ nonzero coefficient, that is, by a monomial transform. ³⁵⁰ Consider the value z_i in the i-th position and some per- ³⁵¹ mutation τ_i of Fq that is linear over Fp. Since φ is also ³⁵² linear over Fp and invertible, we have a bijective linear ³⁵³ transform φ(τ_i(φ⁻¹(·))) of the Simplex code. By (*), there ³⁵⁴ is a monomial automorphismσ_i of the Simplex code whose ³⁵⁵ action on this code is identical toφ(τ_i(φ⁻¹(·))). Then, we have ³⁵⁶ φ(τ_i(z_i)) =σ_i(φ(z_i)), ³⁵⁷ i.e., the action of a Fp-linear permutation τ_i in the i-th ³⁵⁸ coordinate of a word z¯ from Fⁿ_q corresponds to the action ³⁵⁹ of the monomial transformσ_i in thei-th block of coordinates ³⁶⁰

ofφ(¯z). ³⁶¹

Summarizing (i) and (ii), we get the following. If the ³⁶²

mapping ³⁶³

(z₁, . . . , z_n)→

τ_i(z_π−1(1)), . . . , τ_i(z_π−n(n))

364

belongs to AutL_p(C), then ³⁶⁵

(φ(z₁), . . . , φ(z_n))→

σ_i(φ(z_π−1(1))), . . . , σ_i(φ(z_π−n(n)))

366

mapsφ(C)to itself, i.e., belongs toAutM(φ(C)). This proves ³⁶⁷

the statement. ³⁶⁸

Lemma 6. Assume that C is a linear code overFq andC^{⊥ 369}

satisfies the following property: 370

(i) the minimum distance of C^⊥ is at least4. 371

Then the groups AutL_p(C), AutM(φ(C)), and ³⁷²

AutM(φ(C)^⊥)are isomorphic. ³⁷³

Proof: We will first deduce from (i) that ³⁷⁴ (ii) Aut0(φ(C)^⊥) (and hence, also AutM(φ(C)^⊥)) acts ³⁷⁵ imprimitively on the coordinates with the blocks ³⁷⁶

1, . . . ,q−1 p−1

, . . . ,

q−1

p−1(n−1)+1, . . . , q−1 p−1n

. ³⁷⁷

SHIet al.: NEW APPROACH TO THE KASAMI CODES OF TYPE 2 5

(As we see from Lemma 5, this property is indeed necessary,

378

while (i) is only sufficient to guarantee (ii).) To show (ii),

379

we note two other facts. From the code distance ≥4 ofC^⊥,

380

we see that

381

(iii) for every codeword of weight 3 in φ(C)^⊥, the three

382

nonzero coordinates belong to the same block. Indeed,

383

assume that φ(C)^⊥ has a codeword with exactly three

384

nonzero coordinates, with valuesβ,β andβ respec-

385

tively. This means that for somei,i,i,j,j,j such

386

that(i, j)= (i, j)= (i, j)= (i, j), the equation

387

βL(z_iξ^j) +βL(z_iξ^j) +βL(z_iξ^j) = 0

388

is satisfied for every (z₁, . . . , z_n) from C. Since L is

389

linear and nondegenerate, this implies

390

(βξ^j)z_i+ (βξ^j)z_i+ (βξ^j)z_i= 0

391

for all(z₁, . . . , z_n)fromC. The last equation contradicts

392

to the code distance ofC^⊥, unless i=i=i, i.e., the

393

three coordinates are from the same block.

394

On the other hand, because the dual of a Simplex code is a

395

1-perfect Hamming code, we have

396

(iv) every two coordinates from the same block are nonzero

397

coordinates for some weight-3 codeword in φ(C)^⊥.

398

Indeed, in the generator matrix of the Simplex code,

399

the sum of any two columns is proportional to some

400

other column. The same relation takes place between the

401

corresponding coordinates from the same block of the

402

concatenated code, which implies the required property.

403

From (iii) and (iv), we see that the code properties of φ(C)^⊥

404

distinguish the pairs of coordinates in the same block and the

405

pairs of coordinates in different blocks. Hence, the automor-

406

phism group does not break blocks and we have (ii).

407

Next, we observe that property (i) excludes some degenerate

408

cases with vanishing coordinates:

409

(v) every codeword of the Simplex code occurs in every

410

block, over the codewords of the concatenated code

411

φ(C).Indeed, this obviously equivalent to the fact that

412

for every symbol of Fq and every position, there is

413

a codeword of C with the given symbol in the given

414

position. SinceC is linear, it is the same as to say that

415

there is no vanishing coordinate inC. Equivalently,C^⊥

416

does not contain a weight-1 codeword, which follows

417

immediately from (i).

418

Now, we have that every automorphism from

419

AutM(φ(C)^⊥) permutes blocks, permutes coordinates

420

inside each block, and permutes the values of Fp for every

421

coordinate (fixing 0). The same is true for AutM(φ(C)),

422

because it is isomorphic to AutM(φ(C)^⊥) with the

423

isomorphism keeping the permutation part. According to

424

(v), the last two steps (permuting coordinates inside each

425

block, and permuting the values of F_p for every coordinate)

426

must be compatible with the automorphism group of the

427

Simplex code, and result in a permutation of its codewords.

428

A permutation of the codewords of the Simplex code in each

429

block, for the codeφ(C), corresponds to a permutation of the

430

elements of Fq in the corresponding coordinate for C; and

431

a permutation of the blocks corresponds to a permutation of

432

the coordinates for C. So, we see that every automorphism ⁴³³ fromAutM(φ(C))corresponds to some automorphism ofC: ⁴³⁴

AutM(φ(C))Aut0(C). ⁴³⁵

Moreover, because all considered transformations are linear ⁴³⁶

overFp, we have ⁴³⁷

AutM(φ(C))AutL_p(C). ⁴³⁸ From Lemma 5, we have the inverse correspondence. It ⁴³⁹ remains to note that AutM of a code and its dual are ⁴⁴⁰

isomorphic. ⁴⁴¹

IV. CLASSICALKASAMICODES 442

Let q = 2^m for some integerm≥ 1. Consider the cyclic ⁴⁴³ code M_q^⊥ of length q+ 1 defined over Fq by the check ⁴⁴⁴ polynomial h(x) = (x−1)(x−ζ)(x−ζ^q), for the root ⁴⁴⁵ ζ = α^q−1 of order q+ 1 over Fq², where α is a primitive ⁴⁴⁶ root of Fq² (for some notation reasons, we first define M_q^{⊥ 447} and thenM_q as the dual toM_q^⊥). Note thatζ^q2=ζ, so that ⁴⁴⁸ (x−ζ)(x−ζ^q) =x²−(ζ+ζ^q)x+ζ^q+1 =x²−(Tr_q2/qζ)x+1 449

has its coefficients inFq. ⁴⁵⁰

Theorem 1. The codeM_q^⊥is MDS of parameters[q+1,3, q− ⁴⁵¹

1]_q. Its weight distribution is ⁴⁵²

0,1 ,

q−1,q³−q 2

,

q, q²−1 ,

q+1,q³−2q²+q 2

. ⁴⁵³ (4) ⁴⁵⁴

Proof: The BCH bound (see e.g. [13, § 7.6]) applied to ⁴⁵⁵ M_q shows that M_q, hence M_q^⊥ is MDS. The frequency of ⁴⁵⁶ weightq−1is derived on applying [13, Corollary 5, p. 320]. ⁴⁵⁷

A_q−1= (q−1) q+ 1

q−1

= (q−1) q+ 1

2

. ⁴⁵⁸

The frequency of weightqis derived on applying [13, p. 320]. 459

A_q= q+ 1

3−2

(q²−1)−q(q−1)

= (q+ 1)(q−1). ⁴⁶⁰

The frequency of weightq+ 1is derived by complementation ⁴⁶¹

A_q+1=q³−1−A_q−1−A_q. ⁴⁶²

Denote by S_q the binary Simplex code of parameters ⁴⁶³ [q−1, m,^q₂]₂. The main result of this section is the following ⁴⁶⁴

theorem. ⁴⁶⁵

Theorem 2. The concatenation of M_q^⊥ with S_q is a binary ⁴⁶⁶ cyclic codeK_q^⊥ of parameters[q²−1,3m], with nonzerosα, ⁴⁶⁷ α^1+q, withαa primitive root ofF_q². ⁴⁶⁸ Proof: As ζ=α^q−1 is a root of orderq+ 1of F_q², by ⁴⁶⁹

the facts of Section 2.4, we have ⁴⁷⁰

M_q^⊥ = c+ Tr_q²_/q(δζ^j)_q+1

j=1 c∈Fq, δ∈Fq²

. ⁴⁷¹

Let θ=α^1+q. Clearly θ is a root of orderq−1 ofFq. The 472

concatenation mapφfromFq toS_q is conveniently described 473

by using the trace functionTr_q/2. For allβ∈Fq let ⁴⁷⁴ φ(β) =

Tr_q/2(βθ),Tr_q/2(βθ²), . . . ,Tr_q/2(βθ^q−1)

475

=

Tr_q/2(βθⁱ)_q−1

i=1. ⁴⁷⁶