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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 length****q**^{2}**−****1, with**

1

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

2

**code of length** **q****+ 1** **over** **F****q****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 related****q-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 2^{2m}*−*1 with the two zeros *α,α*^{1+2}* ^{m}*,

29

where *α* denotes a primitive root of F_{2}^{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−1}^{2}* ^{−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 zeros

*α,α*

^{(p−1)(1+q)}, where

^{40}

*α*denotes a primitive root of F

_{q}^{2}. 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}
resulting*p-ary (p*is 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 alphabet

*p*such that

*q*is 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 graphs

*H(n, q)*(in

^{72}the role of the

*cover) and*

*H*(

_{p−1}

^{q−1}*n, 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 vertex*v*¯is the set of vertices connected to*v.*¯

102

The*degree*of a vertex¯*v*is the size ofΓ(¯*v). A graph isregular*

103

if every vertex has the same degree. The*i-neighborhood*Γ* _{i}*(¯

*v)*

104

is the set of vertices at geodetic distance *i* to*v. A graph is*¯

105

*distance regular*(DR) if for every pair or vertices *u*¯and*v*¯at

106

distance *i*apart the quantities

107

*b** _{i}*=

*|Γ*

*i+1*(¯

*u)∩*Γ(¯

*v)|,*

*c*

*=*

_{i}*|Γ*

*i−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* F^{n}* _{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 F

*q*; i.e.,

*σ◦*

*π:*

118

(v_{1}*, . . . , v** _{n}*)

*→*(σ

_{1}(v

_{π}*−1*(1)), . . . , σ

*(v*

_{n}

_{π}*−1*(n))).

119

*B. Equitable Partitions, Completely Regular Codes*

120

A partition (P_{0}*, . . . , P** _{k}*) of the vertex set of a graph Γ is

121

called an*equitable 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 every

*v*¯

*∈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}*as the*

_{i,j=0}*intersec-*

127

*tion* (or quotient) *matrix.* If the intersection matrix

128

is tridiagonal, then *P*_{0} is called a *completely regu-*

129

*lar code* of covering radius *k* and *intersection array*

130

(S_{01}*, S*_{12}*, . . . , S*_{k−1}* _{k}*;

*S*

_{10}

*, S*

_{21}

*, . . . , S*

*). That is to say,*

_{k k−1}131

a set of vertices*C*is 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,...*of

*C*with respect to a vertex

*v*¯depends

137

only on the distance from*v*¯to*C.*

138

*C. Linear and Additive Codes* ^{139}

A linear code (that is, a linear subspace ofF^{n}* _{q}*) of length

*n,*

^{140}dimension

*k, minimum distanced*over the fieldF

*q*is called

^{141}an [n, k, d]

*code. An[n, k, d]*

_{q}*code is called an MDS code*

_{q}^{142}if

*d*=

*n−k*+ 1. The duality is understood with respect

^{143}to the standard inner product. Let

*A*

*denote the number of*

_{w}^{144}

codewords of weight*w.* ^{145}

Linear codes are partial cases of the additive codes, which ^{146}
are, by definition, the subgroups of the additive group ofF^{n}* _{q}*. A

^{147}

*coset*of an additive code

*C*is any translate of

*C*by a constant

^{148}vector. A

*coset leader*is any coset element that minimizes

^{149}the weight. The

*weight of a coset*is the weight of any of its

^{150}leaders. The

*coset graph*Γ

*of an additive code*

_{C}*C*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* *{b*0*, . . . , b** _{ρ−1}*;

*c*

_{1}

*, . . . , c*

_{ρ}*}*

*if and only if the*

^{156}

*coset graph is distance-regular with intersection numbers*

^{157}

*b*

_{0}

*, . . . , b*

_{ρ−1}*, c*

_{1}

*, . . . , 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 weight*w.* ^{161}

*D. Cyclic Codes* ^{162}

A cyclic code*C* of length*n* overF*q* is, up to polynomial ^{163}
representation of vectors, an ideal of the ringF*q*[x]/(x^{n}*−*1). ^{164}
All ideals are principal with a unique monic generator, called ^{165}
*the generator polynomial*of*C, and denoted by* *g(x). If this* ^{166}
polynomial has*t*irreducible factors overF*q*, then the code*C* ^{167}
is called a cyclic code*withtzeros. In that case, the dualC*^{⊥}^{168}
admits a trace representation with*t* terms of the form ^{169}

*c(a*_{1}*, . . . , a** _{t}*) =

^{t}*i=1*

Tr(a_{i}*α*^{j}* _{i}*)

_{n}*j=1**,* ^{170}

where ^{171}

*•* *k** _{i}* is the degree of the factor indexed

*i*of

*g(x),*

^{172}

*•* Tr = Tr_{q}*ki**/q* is the trace fromF_{q}*ki* down toF*q*, ^{173}

*•* *a** _{i}* is arbitrary inF

_{q}*ki*,

^{174}

*•* *α** _{i}* is a primitive root inF

^{×}

_{q}*,*

_{ki}^{175}In general the

*relative trace*from F

*q*

*to F*

^{r}*q*, where

*q*is a

^{176}

power of a prime, is defined as ^{177}

Tr_{q}*r**/q*(z) =*z*+*z** ^{q}*+

*· · ·*+

*z*

^{q}

^{r−1}*.*

^{178}If

*q*is prime, then Tr

_{q}*r*

*/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}
length*n*is the stabilizer of*C* 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}

SHI*et al.: NEW APPROACH TO THE KASAMI CODES OF TYPE 2* 3

We will say that a subgroup*A* 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* *N*_{1}, …, *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). The*monomial automorphism group* AutM(C) of a

198

*q-ary codeC* is defined as the set of all*monomial transforms*

199

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

200

form acting onF^{n}* _{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 the*semilinear 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

automorphism*τ* of the fieldF*q* (acting simultaneously on the

208

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

209

for any linear code*C, the groups*AutM(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 F*q* has a subfield F*p*, we also use

213

the notation AutL* _{p}*(C) to denote the subgroup of Aut0(C)

214

consisting of only transformations that are linear overF*p*. Any

215

automorphism from AutL* _{p}*(C) acts as the composition of

216

a coordinate permutation and a F*p*-linear permutation (from

217

GL(log_{p}*q, p)) of* F*q* in every coordinate. In particular, if *q*

218

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

219

AutL_{2}(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 the*inner 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 F

*q*. We assume that

*k*and

*p−*1 are

232

coprime; hence, so are ^{q−1}* _{p−1}* and

*p−*1 (indeed, by induction,

233

*p*^{k}*−1*

*p−1* *≡k*mod*p−*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 of*ξ*are mutually linear indepen-

236

dent overF*p*. Then, the matrix(ξ^{1}*, . . . , ξ*^{q−1}* ^{p−1}*), considered as

237

a*k×*^{q−1}* _{p−1}* matrix overF

*p*, is a generator matrix of a Simplex

238

code of size*q, 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}
F*p*-linear function fromF*q* toF*p*. Any such function can be ^{242}
represented as*L(·*)*≡*Tr* _{q/p}*(a

*·*) for some

*a∈*F

*q*, and in our 243

context we can always assume that *L* is Tr* _{q/p}*. Define

*φ*: 244

F*q* *→*F*p*^{q−1}* ^{p−1}* by

^{245}

*φ(z) =*

*L(zξ*^{1}), . . . , L(zξ^{q−1}* ^{p−1}*)

; ^{246}

expand its action toF^{n}* _{q}* coordinatewise, and to any set of words

^{247}

inF^{n}* _{q}* wordwise.

^{248}

**Lemma 2.** *For any two words* *y,*¯ *z*¯ *in* F^{n}_{q}*, we have* ^{249}
*d(φ(¯y), φ(¯z)) =p*^{k−1}*d(¯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 weight*p** ^{k−1}*, we have

^{252}

*d(φ(y*

_{1}

*, . . . , y*

*), φ(z*

_{n}_{1}

*, . . . , z*

*))*

_{n}^{253}

=*d*

(φ(y_{1}), . . . , φ(y* _{n}*)),(φ(z

_{1}), . . . , φ(z

*))*

_{n}254

=
*n*
*i=1*

*d(φ(y** _{i}*), φ(y

*)) =*

_{i}*n*

*i=1*

*p*^{k−1}*d(y*_{i}*, y** _{i}*)

^{255}

=*p*^{k−1}*d((y*_{1}*, . . . , y** _{n}*),(z

_{1}

*, . . . , z*

*)).*

_{n}^{256}

257

**Lemma 3.** *Assume that* *M* = (M* _{ij}*)

^{m}

_{i=1}

^{n}

_{j=1}*is an*

*m×n*

^{258}

*generator matrix of a codeC*

*in*F

^{n}

_{q}*. Then the matrix*

^{259}

Φ(M) =

⎛

⎜⎝

*M*_{11}*ξ*^{1} *... M*_{11}*ξ*^{q−1}^{p−1}*M*_{12}*ξ*^{1} *. . . M*_{1n}*ξ*^{p−1}^{q−1}*...* *...* *...* *...* *. . .* *...*

*M*_{m1}*ξ*^{1} *... M*_{m1}*ξ*^{q−1}^{p−1}*M*_{m2}*ξ*^{1} *. . . M*_{mn}*ξ*^{q−1}^{p−1}

⎞

⎟⎠*,* ^{260}

*considered as a* *km×* ^{q−1}_{p−1}*n* *matrix over* F*p**, is a generator* ^{261}
*matrix of the codeφ(C)* *in*F*p*^{p−1}^{q−1}^{n}*.* ^{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

_{i1}*ξ*

^{1}), . . . , L

*(M*

_{i}

_{in}*ξ*

^{q−1}*)*

^{p−1}266

for some F*p*-linear functions *L** _{i}* : F

*q*

*→*F

*p*. Since

*L*

*(*

_{i}*·*)

*≡*

^{267}

*L(a*

_{i}*·*)for some

*a*

_{i}*∈*F

*q*, we have

^{268}

¯
*c* =

*m*
*i=1*

*L(a*_{i}*M*_{i1}*ξ*^{1}), . . . , L(a_{i}*M*_{in}*ξ*^{p−1}* ^{q−1}*)

269

=
*m*
*i=1*

*φ(a*_{i}*M** _{i1}*), . . . , φ(a

_{i}*M*

*)*

_{in}270

=
*m*
*i=1*

*φ(a*_{i}*M*_{i1}*, . . . , a*_{i}*M** _{in}*)

271

= *φ*
^{m}

*i=1*

*a** _{i}*(M

_{i1}*, . . . , M*

*)*

_{in}*.* ^{272}

That is,*c*¯is a*φ-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* *in*F^{n}_{q}*and the coset graph*Γ_{φ(C)}*⊥* *of the dualφ(C)*^{⊥}

278

*of its imageφ(C)* *in*F*p*^{p−1}^{q−1}^{n}*are isomorphic.*

279

*Proof:* We first observe that the number of cosets of*C*^{⊥}

280

coincides with size of the code*C, which is the same as the size*

281

of the code*φ(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

Let*C*be a code overF*q* with a generator matrix*M*, 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 matrix*M* = (M* _{ij}*)

^{m}

_{i=1}

^{n}*over*

_{j=1}295

F*q*, the connecting syndromes are

296

*a(M*_{1j}*, . . . , M** _{mj}*)

^{T}

*,*

*j∈ {1, . . . , n},*

*a∈*F

^{×}*(2)*

_{q}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 for*φ(C)*and a check matrix for*φ(C)** ^{⊥}*. The

301

connecting syndromes are

302

*b(ξ*^{l}*M*_{1j}*, . . . , ξ*^{l}*M** _{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 set*S.*

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 group*AutL* _{p}*(C)

*is isomorphic to a subgroup*

322

*of*AutM(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−*1*n*

*.*

325

*Proof:* The action of the monomial automorphism group ^{326}
on a codeword*c*of*C* can be represented as a permutation of ^{327}
the coordinates and, for every *i*from 1 to*n*the 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 for*φ(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 from*φ(z*_{1}*, . . . , z** _{n}*)by sending each

*i-th block of*

^{335}coordinates to the place of the

*φ(i)-th block.*

^{336}(ii) Next, we show that the action of aF

*p*-linear permutation

^{337}on the value

*z*

*in the*

_{i}*i-th coordinate corresponds to a*

^{338}monomial transform inside the

*i-th block of*

*φ(z*

_{1}

*, . . . , 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 height

*m. 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*

*in the*

_{i}*i-th position and some per-*

^{351}mutation

*τ*

*of F*

_{i}*q*that is linear over F

*p*. Since

*φ*is also

^{352}linear over F

*p*and invertible, we have a bijective linear

^{353}transform

*φ(τ*

*(φ*

_{i}*(·))) of the Simplex code. By (*), there*

^{−1}^{354}is a monomial automorphism

*σ*

*of the Simplex code whose*

_{i}^{355}action on this code is identical to

*φ(τ*

*(φ*

_{i}*(·))). Then, we have*

^{−1}^{356}

*φ(τ*

*(z*

_{i}*)) =*

_{i}*σ*

*(φ(z*

_{i}*)),*

_{i}^{357}i.e., the action of a F

*p*-linear permutation

*τ*

*in the*

_{i}*i-th*

^{358}coordinate of a word

*z*¯ from F

^{n}*corresponds to the action*

_{q}^{359}of the monomial transform

*σ*

*in the*

_{i}*i-th block of coordinates*

^{360}

of*φ(¯z).* ^{361}

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

mapping ^{363}

(z_{1}*, . . . , z** _{n}*)

*→*

*τ** _{i}*(z

_{π}*−1*(1)), . . . , τ

*(z*

_{i}

_{π}*−n*(n))

364

belongs to AutL* _{p}*(C), then

^{365}

(φ(z_{1}), . . . , φ(z* _{n}*))

*→*

*σ** _{i}*(φ(z

_{π}*−1*(1))), . . . , σ

*(φ(z*

_{i}

_{π}*−n*(n)))

366

maps*φ(C)*to itself, i.e., belongs toAutM(φ(C)). This proves ^{367}

the statement. ^{368}

**Lemma 6.** *Assume that* *C* *is a linear code over*F*q* *andC*^{⊥}^{369}

*satisfies the following property:* 370

(i) *the minimum distance of* *C*^{⊥}*is at least*4. 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−*1*n*

*.* ^{377}

SHI*et 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 of*C** ^{⊥}*,

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 values*β*,*β** ^{}* and

*β*

*respec-*

^{}385

tively. This means that for some*i,i** ^{}*,

*i*

*,*

^{}*j,j*

*,*

^{}*j*

*such*

^{}386

that(i, j)*= (i*^{}*, j** ^{}*)

*= (i*

^{}*, j*

*)*

^{}*= (i, j), the equation*

387

*βL(z*_{i}*ξ** ^{j}*) +

*β*

^{}*L(z*

_{i}

^{}*ξ*

^{j}*) +*

^{}*β*

^{}*L(z*

_{i}

^{}*ξ*

^{j}*) = 0*

^{}388

is satisfied for every (z_{1}*, . . . , z** _{n}*) from

*C. Since*

*L*is

389

linear and nondegenerate, this implies

390

(βξ* ^{j}*)z

*+ (β*

_{i}

^{}*ξ*

^{j}*)z*

^{}*+ (β*

_{i}

^{}*ξ*

^{j}*)z*

^{}*= 0*

_{i}391

for all(z_{1}*, . . . , z** _{n}*)from

*C. The last equation contradicts*

392

to the code distance of*C** ^{⊥}*, 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 F*q* and every position, there is

413

a codeword of *C* with the given symbol in the given

414

position. Since*C* is linear, it is the same as to say that

415

there is no vanishing coordinate in*C. 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 F*p* 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 code*φ(C), corresponds to a permutation of the*

430

elements of F*q* 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 of*C:* ^{434}

AutM(φ(C))Aut0(C). ^{435}

Moreover, because all considered transformations are linear ^{436}

overF*p*, 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 integer

*m≥*1. Consider the cyclic

^{443}code

*M*

_{q}*of length*

^{⊥}*q*+ 1 defined over F

*q*by the check

^{444}polynomial

*h(x) = (x−*1)(x

*−ζ)(x−ζ*

*), for the root*

^{q}^{445}

*ζ*=

*α*

*of order*

^{q−1}*q*+ 1 over F

*q*

^{2}, where

*α*is a primitive

^{446}root of F

*q*

^{2}(for some notation reasons, we first define

*M*

_{q}

^{⊥}^{447}and then

*M*

*as the dual to*

_{q}*M*

_{q}*). Note that*

^{⊥}*ζ*

^{q}^{2}=

*ζ, so that*

^{448}(x

*−ζ)(x−ζ*

*) =*

^{q}*x*

^{2}

*−*(ζ+ζ

*)x+ζ*

^{q}*=*

^{q+1}*x*

^{2}

*−*(Tr

*2*

_{q}*/q*

*ζ)x+1*449

has its coefficients inF*q*. ^{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,*q*^{3}*−q*
2

*,*

*q, q*^{2}*−1*
*,*

*q+1,q*^{3}*−*2q^{2}+*q*
2

*.* ^{453}
(4) ^{454}

*Proof:* The BCH bound (see e.g. [13, § 7.6]) applied to ^{455}
*M** _{q}* shows that

*M*

*, hence*

_{q}*M*

_{q}*is MDS. The frequency of*

^{⊥}^{456}weight

*q−*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 weight*q*is derived on applying [13, p. 320]. 459

*A** _{q}*=

*q*+ 1

3*−*2

(q^{2}*−*1)*−q(q−*1)

= (q+ 1)(q*−*1). ^{460}

The frequency of weight*q*+ 1is derived by complementation ^{461}

*A** _{q+1}*=

*q*

^{3}

*−*1

*−A*

_{q−1}*−A*

_{q}*.*

^{462}

Denote by *S** _{q}* the binary Simplex code of parameters

^{463}[q

*−*1, m,

^{q}_{2}]

_{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*[q^{2}*−*1,3m], with nonzeros*α,* ^{467}
*α*^{1+q}*, withαa primitive root of*F_{q}^{2}*.* ^{468}
*Proof:* As *ζ*=*α** ^{q−1}* is a root of order

*q*+ 1of F

_{q}^{2}, by

^{469}

the facts of Section 2.4, we have ^{470}

*M*_{q}* ^{⊥}* =

*c*+ Tr

_{q}^{2}

*(δζ*

_{/q}*)*

^{j}

_{q+1}*j=1* *c∈*F*q**, δ∈*F*q*^{2}

*.* ^{471}

Let *θ*=*α*^{1+q}. Clearly *θ* is a root of order*q−*1 ofF*q*. The 472

concatenation map*φ*fromF*q* to*S** _{q}* is conveniently described 473

by using the trace functionTr* _{q/2}*. For all

*β∈*F

*q*let

^{474}

*φ(β*) =

Tr* _{q/2}*(βθ),Tr

*(βθ*

_{q/2}^{2}), . . . ,Tr

*(βθ*

_{q/2}*)*

^{q−1}475

=

Tr* _{q/2}*(βθ

*)*

^{i}

_{q−1}*i=1**.* ^{476}