几类二重和三重线性码的构造

杜小妮,李晓丹,吕红霞,赵丽萍

(西北师范大学 数学与统计学院,甘肃 兰州 730070)

0 引言

文中假设p为奇素数,q=pm,m为正整数.由有限域Fq到Fp的迹函数[9]Trm(·) 定义为:

Trm(α)=α+αp+…+αpm-1, ∀α∈Fpm.

设F*q表示Fq中全体非零元素组成的集合,集合D={d1,d2,…,dn}⊆Fq,则Fq上长度为n的线性码定义为

称集合D为线性码CD的定义集.通过选择合适的定义集D可以构造一些较低重量的线性码[5].近年来,通过选择不同的定义集得到了几类较低重量的线性码[10-18].研究表明,恰当地选择定义集可以得到一些最佳码[8,19-20].

若m≥2为正整数,Li等[7]通过选取

构造了p元线性码

CD={c(a,b):a,b∈Fpm},

(1)

得到了几类二重和三重的线性码.其中

受文献[7]的启发,本文选择定义集

其中c∈F*p,l为正整数且l∈{1,2,pm/2+1},m≥2为正整数.下面讨论由该定义集构造的几类线性码的重量分布.

1 基础知识

首先给出指数和的一些结论以及证明主要结论需要用到的引理.

对任意的a∈Fq,Fq上的加法特征定义为

称Fq上乘法群F*q的特征为Fq的乘法特征[9],定义为

其中g是F*q的一个生成元.补充定义λj(0)=0.称乘法特征λ(q-1)/2为Fq的二次特征,用η来表示.

引理2[22-23]设λ为F*q上的一个N>2阶乘法特征.假设存在最小正整数f使得pf≡-1(modN).若m=2ft,t为某个正整数,则对1≤i≤N-1,有

引理3[9]设λ为Fq上阶为N=gcd(n,q-1)≥2的一个乘法特征,则对任意的a∈F*q,有

引理4[9]若f(x)=a2x2+a1x+a0∈Fq[x],其中a2≠0,则

引理5[24]设m=2s(s为正整数),a∈F*ps,b∈Fpm,则

引理6[6,13]对每个c∈F*p,有

引理7对每个c∈F*p,设

Mc={b∈F*q:Trm(bps+1)=c},

则|Mc|=pm-1+pm/2-1.

证明由引理2和引理3可得

所以码长n=p2m-1-1.

则码字的重量

W(c(a,b))=n-N(a,b),

(4)

且有

其中,

2.1 l=1的情形

表1 码的重量分布

证明由(5)式可以得到

因而,依据(4)式可得到定理的结论. 】

2.2 l=2的情形

表2 m为偶数时的重量分布

证明分以下4种情况来确定N(a,b)的值.

( i )若a=b=0,则由(5)式可得

由引理6可知该值出现的次数为

(iv)若a∈FqFp,b∈Fq或a=0,b∈F*q,则

显然该值出现的次数为(q-p)q+q-1,即

pm(pm-p+1)-1.

由(4)式可得到码的重量分布. 】

证明分以下3种情况来确定N(a,b)的值.

( i )若a=b=0,则

由引理6可知该值出现的次数为

由(4)式可定义

其对应的重数分别为Aw1,Aw2,Aw3,根据MacWilliams方程[14]可得

解方程可得该码的重量分布. 】

2.3 l=pm/2+1的情形

令m=2s(s为一个整数).与l=2的情形类似可知,若a∉F*p则Ω3=0.若a∈F*p则由引理5有

表的重量分布

证明依据引理7,该定理的证明方法与定理2的类似,此处不再赘述. 】

3 结束语

根据文献[6]的结论,文中构造的线性码可应用于秘密共享方案.

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