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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 lengthq21, 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

ISTANCE-REGULAR graphs form the most extensively

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 22m1 with the two zeros α,α1+2m,

29

where α denotes a primitive root of F22m. 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 qp−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 Fq2. 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 = pk, 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−1q−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/

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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-neighborhoodΓiv)

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

bi=i+1u)∩Γ(¯v)|, ci=i−1u)∩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 Fnq, 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 σi of Fq; i.e., σ◦ π:

118

(v1, . . . , vn)1(vπ−1(1)), . . . , σn(vπ−1(n))).

119

B. Equitable Partitions, Completely Regular Codes

120

A partition (P0, . . . , Pk) 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 Sij

123

such that

124

|Γ(¯v)∩Pj|=Sij for everyv¯∈Pi, i, j= 0, . . . , k.

125

The numbers Sij are referred to as the intersection

126

numbers, and the matrix (Sij)ki,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, . . . , Sk−1k;S10, S21, . . . , Sk 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

(|CΓiv)|)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 ofFnq) 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 Aw 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 ofFnq. 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 graphΓC 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, . . . , bρ−1;c1, . . . , cρ} if and only if the 156 coset graph is distance-regular with intersection numbers 157 b0, . . . , bρ−1, c1, . . . , cρ. 158 The weight distribution of a code is displayed as 159

[0,1,· · ·,w, Aw,· · ·], where Aw 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]/(xn1). 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, . . . , at) = t

i=1

Tr(aiαji) n

j=1, 170

where 171

ki is the degree of the factor indexediofg(x), 172

Tr = Trqki/q is the trace fromFqki down toFq, 173

ai is arbitrary inFqki, 174

αi is a primitive root inF×qki, 175 In general the relative trace from Fqr to Fq, where q is a 176

power of a prime, is defined as 177

Trqr/q(z) =z+zq+· · ·+zqr−1. 178 If q is prime, then Trqr/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

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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, …, Nt, 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 onFnq is the product of a permutation matrix by

201

an invertible diagonal matrix, both in Fn×nq . 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 AutLp(C) to denote the subgroup of Aut0(C)

214

consisting of only transformations that are linear overFp. Any

215

automorphism from AutLp(C) acts as the composition of

216

a coordinate permutation and a Fp-linear permutation (from

217

GL(logpq, 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 = pk. Let ξ be a root of

231

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

232

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

233

pk−1

p−1 ≡kmodp−1). It follows that

234

F×q =

l|b∈F×p, l∈ {1, . . . ,q−1p−1}

(1)

235

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

236

dent overFp. Then, the matrix(ξ1, . . . , ξq−1p−1), considered as

237

aq−1p−1 matrix overFp, is a generator matrix of a Simplex

238

code of sizeq, where the distance between any two different

239

codewords is pk−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(·)Trq/p(a·) for somea∈Fq, and in our 243

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

Fq Fpq−1p−1 by 245

φ(z) =

L(zξ1), . . . , L(zξq−1p−1)

; 246

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

inFnq wordwise. 248

Lemma 2. For any two words y,¯ z¯ in Fnq, we have 249 d(φ(¯y), φ(¯z)) =pk−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 weightpk−1, we have 252 d(φ(y1, . . . , yn), φ(z1, . . . , zn)) 253

=d

(φ(y1), . . . , φ(yn)),(φ(z1), . . . , φ(zn))

254

= n i=1

d(φ(yi), φ(yi)) = n i=1

pk−1d(yi, yi) 255

=pk−1d((y1, . . . , yn),(z1, . . . , zn)). 256

257

Lemma 3. Assume that M = (Mij)mi=1nj=1 is an m×n 258 generator matrix of a codeC inFnq. Then the matrix 259

Φ(M) =

⎜⎝

M11ξ1 ... M11ξq−1p−1 M12ξ1 . . . M1nξp−1q−1 ... ... ... ... . . . ...

Mm1ξ1 ... Mm1ξq−1p−1 Mm2ξ1 . . . Mmnξq−1p−1

⎟⎠, 260

considered as a km× q−1p−1n matrix over Fp, is a generator 261 matrix of the codeφ(C) inFpp−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

Li(Mi1ξ1), . . . , Li(Minξq−1p−1)

266

for some Fp-linear functions Li : Fq Fp. SinceLi(·) 267 L(ai·)for someaiFq, we have 268

¯ c =

m i=1

L(aiMi1ξ1), . . . , L(aiMinξp−1q−1)

269

= m i=1

φ(aiMi1), . . . , φ(aiMin)

270

= m i=1

φ(aiMi1, . . . , aiMin)

271

= φ m

i=1

ai(Mi1, . . . , Min)

. 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

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Lemma 4. The coset graph ΓC of the dualC of a linear

277

codeC inFnq and the coset graphΓφ(C) of the dualφ(C)

278

of its imageφ(C) inFpp−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

ΓC, 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 = (Mij)mi=1nj=1 over

295

Fq, the connecting syndromes are

296

a(M1j, . . . , Mmj)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(ξlM1j, . . . , ξlMmj)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−1q−1 +l. Since {bξl | b F×p, l ∈ {1, . . . ,q−1p−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 ΓC if and only if

309

the corresponding syndromes have the difference in S. Two

310

cosets of φ(C) are adjacent in Γφ(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 ΓC andΓφ(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 groupAutLp(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 ofFmp . 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(zπ−1(1), . . . , zπ−n(n)) = (φ(zπ−1(1)), . . . , φ(zπ−n(n))) 334 is obtained fromφ(z1, . . . , zn)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 zi in the i-th coordinate corresponds to a 338 monomial transform inside the i-th block of φ(z1, . . . , zn). 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 zi in the i-th position and some per- 351 mutation τi 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(τi−1(·))) of the Simplex code. By (*), there 354 is a monomial automorphismσi of the Simplex code whose 355 action on this code is identical toφ(τi−1(·))). Then, we have 356 φ(τi(zi)) =σi(φ(zi)), 357 i.e., the action of a Fp-linear permutation τi in the i-th 358 coordinate of a word z¯ from Fnq corresponds to the action 359 of the monomial transformσi in thei-th block of coordinates 360

ofφ(¯z). 361

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

mapping 363

(z1, . . . , zn)

τi(zπ−1(1)), . . . , τi(zπ−n(n))

364

belongs to AutLp(C), then 365

(φ(z1), . . . , φ(zn))

σi(φ(zπ−1(1))), . . . , σi(φ(zπ−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 AutLp(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

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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(ziξj) +βL(ziξj) +βL(ziξj) = 0

388

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

389

linear and nondegenerate, this implies

390

(βξj)zi+ (βξj)zi+ (βξj)zi= 0

391

for all(z1, . . . , zn)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 Fp 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))AutLp(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 = 2m for some integerm≥ 1. Consider the cyclic 443 code Mq of length q+ 1 defined over Fq by the check 444 polynomial h(x) = (x−1)(x−ζ)(x−ζq), for the root 445 ζ = αq−1 of order q+ 1 over Fq2, where α is a primitive 446 root of Fq2 (for some notation reasons, we first define Mq 447 and thenMq as the dual toMq). Note thatζq2=ζ, so that 448 (x−ζ)(x−ζq) =x2(ζ+ζq)x+ζq+1 =x2(Trq2/qζ)x+1 449

has its coefficients inFq. 450

Theorem 1. The codeMqis 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,q32q2+q 2

. 453 (4) 454

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

Aq−1= (q1) q+ 1

q−1

= (q1) q+ 1

2

. 458

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

Aq= q+ 1

32

(q21)−q(q−1)

= (q+ 1)(q1). 460

The frequency of weightq+ 1is derived by complementation 461

Aq+1=q31−Aq−1−Aq. 462

Denote by Sq the binary Simplex code of parameters 463 [q1, m,q2]2. The main result of this section is the following 464

theorem. 465

Theorem 2. The concatenation of Mq with Sq is a binary 466 cyclic codeKq of parameters[q21,3m], with nonzerosα, 467 α1+q, withαa primitive root ofFq2. 468 Proof: As ζ=αq−1 is a root of orderq+ 1of Fq2, by 469

the facts of Section 2.4, we have 470

Mq = c+ Trq2/q(δζj)q+1

j=1 c∈Fq, δ∈Fq2

. 471

Let θ=α1+q. Clearly θ is a root of orderq−1 ofFq. The 472

concatenation mapφfromFq toSq is conveniently described 473

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

Trq/2(βθ),Trq/2(βθ2), . . . ,Trq/2(βθq−1)

475

=

Trq/2(βθi)q−1

i=1. 476

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