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Optimal Self-Dual Codes over F2+ vF2

Koichi Betsumiya

Graduate School of Mathematics

Nagoya University

Nagoya 464–8602, Japan

and

Masaaki Harada

Department of Mathematical Sciences

Yamagata University

Yamagata 990–8560, Japan

October 19, 2000

Abstract

In this correspondence, we study optimal self-dual codes and Type IV self-dual

codes over the ring F2+vF2of order 4. We give improved upper bounds on minimum

Hamming and Lee weights for such codes. Using the bounds, we determine the highest

minimum Hamming and Lee weights for such codes of lengths up to 30. We also

construct optimal self-dual codes and Type IV self-dual codes.

Index Terms: Self-dual codes over rings, Type IV self-dual codes and binary optimal

codes.

1 Introduction

There are four different rings of order 4, namely the finite field F4, the ring Z4of integers

modulo 4 and two other rings denoted by F2+ uF2 = {0,1,u,1 + u} with u2= 0 and

F2+ vF2= {0,1,v,1 + v} with v2= v. Self-dual codes over F4are covered by the classical

coding theory. Recently self-dual codes over Z4and F2+ uF2have been widely studied.

In this correspondence, we study optimal self-dual codes and Type IV self-dual codes

over F2+vF2with respect to the Hermitian and Euclidian inner products. Bachoc [1] studies

Hermitian self-dual codes over F2+ vF2. Upper bounds on the Bachoc minimum weights

of Hermitian self-dual codes are given in [1] and optimal Hermitian self-dual codes with

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respect to the weight are constructed. In this correspondence, we investigate the minimum

Hamming weights and the minimum Lee weights. In [7] and [6], upper bounds on minimum

Hamming and Lee weights of Hermitian self-dual codes and Hermitian Type IV self-dual

codes of lengths up to 24, are given, respectively. In Section 3, we show that the minimum

Hamming weight of a code over F2+ vF2is the same as the minimum Lee weight. Using

some characterization of Hermitian self-dual codes over F2+ vF2, we give improved upper

bounds on minimum Hamming and Lee weights for Hermitian self-dual codes and Hermitian

Type IV self-dual codes and we construct optimal Hermitian self-dual codes and Hermitian

Type IV self-dual codes in Section 4. In Section 5, we investigate optimal Euclidean self-dual

codes.

2Self-Dual Codes over F2+ vF2

A code C of length n over F2+vF2is an (F2+vF2)-submodule of (F2+vF2)nwhere F2+vF2

is a commutative ring {0,1,v,1 + v} with v2= v. An element of C is called a codeword of

C. A generator matrix of C is a matrix whose rows generate C. Three different weights for

codes over F2+ vF2are known, namely the Hamming, Lee and Bachoc weights [1], [6] and

[7]. The Hamming weight of a codeword is the number of non-zero components. The Lee

weights of the elements 0,1,v and 1+v are 0,2,1 and 1, respectively. The Bachoc weight is

defined in [1] and the weights of the elements 0,1,v and 1+v are 0,1,2 and 2, respectively.

The Lee and Bachoc weights of a codeword are the rational sums of the Lee and Bachoc

weights of its components, respectively. The minimum Hamming, Lee and Bachoc weights,

dH, dLand dBof C are the smallest Hamming, Lee and Bachoc weights among all non-zero

codewords of C, respectively.

We define two inner products (x,y) and [x,y] of x and y in (F2+ vF2)nwhere x =

(x1,...,xn) and y = (y1,...,yn) are two elements of (F2+ vF2)n. The Euclidean inner

product is defined as (x,y) = x1y1+···+xnyn, and the Hermitian inner product is defined

as [x,y] = x1y1+···+xnyn, where 0 = 0, 1 = 1, v = v+1 and v + 1 = v. The dual code C⊥

with respect to the Euclidean inner product of C is defined as C⊥= {x ∈ (F2+vF2)n| (x,y) =

0 for all y ∈ C} and the dual code C∗with respect to the Hermitian inner product of C

is defined as C∗= {x ∈ (F2+ vF2)n| [x,y] = 0 for all y ∈ C}. C is Euclidean self-dual if

C = C⊥and C is Hermitian self-dual if C = C∗. Recently Type IV self-dual codes over

rings of order 4 are defined in [6]. A self-dual code is Type IV if all the Hamming weights

are even.

3 Minimum Weights

In this section, we give a characterization of minimum Hamming and Lee weights. We first

study self-dual codes over F2+ vF2using the Chinese remainder theorem.

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Define the map

Φ : F2+ vF2→ F2× F2,

where Φ(0) = (0,0), Φ(1) = (1,1), Φ(v) = (0,1) and Φ(1 + v) = (1,0). Φ is a ring-

isomorphism by the Chinese remainder theorem.

naturally. Let C be a code over F2+ vF2, then there are binary codes C1 and C2 such

that C = Φ−1(C1,C2) and we denote C by CRT(C1,C2). Note that C1and C2are uniquely

determined for each CRT(C1,C2). Using the above map, characterizations of self-dual codes

and Type IV self-dual codes over F2+ vF2are given.

The map is extended to (F2+ vF2)n

Lemma 1 ([5] and [6]) CRT(C1,C2) is a Euclidean self-dual code if and only if C1and

C2 are binary self-dual codes. CRT(C1,C2) is Euclidean Type IV self-dual if and only if

C1= C2.

Lemma 2 ([1] and [6]) CRT(C1,C2) is a Hermitian self-dual code if and only if C2= C⊥

CRT(C1,C⊥

1.

1) is Hermitian Type IV self-dual if and only if C1and C⊥

1are even codes.

Let c be a codeword of C = CRT(C1,C2) then c can be uniquely written as c =

Φ−1(c1,c2) where c1and c2are codewords of C1and C2, respectively. Let wH(c) and wL(c)

be the Hamming and Lee weights of c, respectively. Then

wH(c) = wH(c1) + wH(c2) − wH(c1∗ c2),

wL(c) = wH(c1) + wH(c2),

(1)

where c1∗ c2denotes the Hadamard product of c1and c2.

Proposition 3 Let dHand dLbe the minimum Hamming and Lee weights of CRT(C1,C2),

respectively. Then

dH= dL= min{d(C1),d(C2)},

where d(Ci) denotes the minimum weight of a binary code Ci.

Proof. The two cases are similar, thus we show that dH = min{d(C1),d(C2)}. Let c be

a codeword of CRT(C1,C2) then c = Φ−1(c1,c2) where c1 and c2 are codewords of C1

and C2, respectively. Then it follows from (1) that wH(c) ≥ max{wH(c1),wH(c2)}. Thus

dH≥ min{d(C1),d(C2)}. Assume that d(C1) ≥ d(C2). Let c?

d(C2) in C2then Φ−1(0,c?

2be a codeword with weight

2) is a codeword of Hamming weight d(C2). The result follows. 2

By the above proposition, it is sufficient to consider only minimum Hamming weights.

Thus the minimum Hamming weight is shortly said to be the minimum weight from now

on. We say that a self-dual (resp. Type IV self-dual) code C is optimal if C has the highest

minimum weight among all self-dual (resp. Type IV self-dual) codes of that length.

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4Optimal Hermitian Self-Dual Codes

In [7], weak upper bounds on minimum weights of Hermitian self-dual codes of lengths up

to 24 are given. In this section, we determine the exact highest minimum weight of such

codes for length up to 30.

First we give improved upper bounds on minimum weights using the characterization of

Hermitian self-dual codes given in Section 3.

Proposition 4 Let dmax(n,k) be the highest minimum weight among all binary linear [n,k]

codes. The highest minimum weight dSD(n) among all Hermitian self-dual codes of length n

is bounded by

dSD(n) ≤ dmax(n,?(n + 1)/2?).

Proof. By Lemma 2 and Proposition 3, the minimum weight of a Hermitian self-dual code

CRT(C1,C⊥

1) is min{d(C1),d(C⊥

dSD(n) ≤

1)}. Thus we have

max

1≤k≤n−1min{dmax(n,k),dmax(n,n − k)}.

Since dmax(n,k) ≤ dmax(n,k − 1), the result follows.

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Lower and upper bounds on minimum weights for binary linear codes are given in [3].

Thus one can easily obtain the highest possible minimum weights of Hermitian self-dual

codes using the bounds.

Corollary 5 Let d?

[n,k] codes whose dual codes are even. The highest minimum weight dIV(n) among all

Hermitian Type IV self-dual codes of length n is bounded by

max(n,k) be the highest minimum weight among all binary linear even

dIV(n) ≤ d?

max(n,n/2).

Proof. Similar to that of Proposition 4. Note that a Hermitian Type IV self-dual code of

length n exists if and only if n is even [6].

2

We determine the highest possible minimum weight of Hermitian Type IV self-dual codes

using the following upper bound instead of Corollary 5 since d?

?dSD(n)

max(n,k) is not known.

Corollary 6 dIV(n) ≤ 2

2

?

.

We now present methods to construct optimal Hermitian self-dual codes:

• Method FSD: Let B be a binary formally self-dual [n,n/2,d] code. Note that B⊥is

also an [n,n/2,d] code. By Lemma 2 and Proposition 3, CRT(B,B⊥) is a Hermitian

self-dual code over F2+vF2with minimum weight d of length n. Moreover if B is even

then CRT(B,B⊥) is a Hermitian Type IV self-dual code.

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• Method SD: Let B be a binary self-dual [n,n/2,d] code. Since B and B⊥are even,

CRT(B,B⊥) is a Hermitian Type IV self-dual code with minimum weight d of length

n by Lemma 2.

• Method Type IV: If the existence of a Hermitian Type IV self-dual code with mini-

mum weight d of length n gives one of a Hermitian self-dual code with minimum weight

d of this length. Thus Hermitian Type IV self-dual codes with high minimum weights

given in Table 1 are used to determine the highest minimum weight of Hermitian

self-dual codes (e.g., n = 2,4,8).

• Method P: Let B be a binary formally self-dual [n,n/2,d] code. There is a coordinate

of B such that the punctured code P obtained by deleting the coordinate is an [n −

1,n/2] code. Then it is easy to see that the dual code of P is an [n − 1,n/2 − 1] code

with minimum weight ≥ d − 1. Hence CRT(P,P⊥) is a Hermitian self-dual code of

length n − 1 with minimum weight d or d − 1.

It is well known that there are binary self-dual codes with parameters [2,1,2], [4,2,2],

[6,3,2], [8,4,4], [10,5,4], [16,8,4], [22,11,6], [24,12,8], [26,13,6] and [32,16,8] (cf. [4]).

There are extremal formally self-dual even [n,n/2,2[n/8] + 2] codes of lengths n = 12, 14,

18, 20, 28 and 30 [9]. Thus Hermitian Type IV self-dual codes with the minimum weight

meeting the upper bound given in Corollary 6 are constructed for length up to 32. We list

in Table 1 the exact highest minimum weight dIV(n) of Hermitian Type IV self-dual codes

of length n ≤ 32.

Table 1: The highest minimum weight of Hermitian Type IV self-dual codes of length up to

32

Length n

2

4

6

8

10

12

14

16

dIV(n)

2

2

2

4

4

4

4

4

Method

SD

SD

SD

SD

SD

FSD

FSD

SD

Length n

18

20

22

24

26

28

30

32

dIV(n)

6

6

6

8

6

8

8

8

Method

FSD

FSD

SD

SD

SD

FSD

FSD

SD

Similarly to Hermitian Type IV self-dual codes, by the above methods, we construct

optimal Hermitian self-dual codes with the minimum weight meeting the upper bound given

in Proposition 4 for length n ≤ 32 except n = 11,13,31. We list in Table 2 the exact highest

minimum weight dSD(n) of Hermitian self-dual codes of length n ≤ 30 and n = 32.

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