幾類二重和三重線性碼的構(gòu)造

杜小妮,李曉丹,呂紅霞,趙麗萍

(西北師范大學(xué) 數(shù)學(xué)與統(tǒng)計學(xué)院,甘肅 蘭州 730070)

0 引言

文中假設(shè)p為奇素數(shù),q=pm,m為正整數(shù).由有限域Fq到Fp的跡函數(shù)[9]Trm(·) 定義為:

Trm(α)=α+αp+…+αpm-1, ?α∈Fpm.

設(shè)F*q表示Fq中全體非零元素組成的集合,集合D={d1,d2,…,dn}?Fq,則Fq上長度為n的線性碼定義為

稱集合D為線性碼CD的定義集.通過選擇合適的定義集D可以構(gòu)造一些較低重量的線性碼[5].近年來,通過選擇不同的定義集得到了幾類較低重量的線性碼[10-18].研究表明,恰當(dāng)?shù)剡x擇定義集可以得到一些最佳碼[8,19-20].

若m≥2為正整數(shù),Li等[7]通過選取

構(gòu)造了p元線性碼

CD={c(a,b):a,b∈Fpm},

(1)

得到了幾類二重和三重的線性碼.其中

受文獻[7]的啟發(fā),本文選擇定義集

其中c∈F*p,l為正整數(shù)且l∈{1,2,pm/2+1},m≥2為正整數(shù).下面討論由該定義集構(gòu)造的幾類線性碼的重量分布.

1 基礎(chǔ)知識

首先給出指數(shù)和的一些結(jié)論以及證明主要結(jié)論需要用到的引理.

對任意的a∈Fq,Fq上的加法特征定義為

稱Fq上乘法群F*q的特征為Fq的乘法特征[9],定義為

其中g(shù)是F*q的一個生成元.補充定義λj(0)=0.稱乘法特征λ(q-1)/2為Fq的二次特征,用η來表示.

引理2[22-23]設(shè)λ為F*q上的一個N>2階乘法特征.假設(shè)存在最小正整數(shù)f使得pf≡-1(modN).若m=2ft,t為某個正整數(shù),則對1≤i≤N-1,有

引理3[9]設(shè)λ為Fq上階為N=gcd(n,q-1)≥2的一個乘法特征,則對任意的a∈F*q,有

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

引理5[24]設(shè)m=2s(s為正整數(shù)),a∈F*ps,b∈Fpm,則

引理6[6,13]對每個c∈F*p,有

引理7對每個c∈F*p,設(shè)

Mc={b∈F*q:Trm(bps+1)=c},

則|Mc|=pm-1+pm/2-1.

證明由引理2和引理3可得

所以碼長n=p2m-1-1.

則碼字的重量

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

(4)

且有

其中,

2.1 l=1的情形

表1 碼的重量分布

證明由(5)式可以得到

因而,依據(jù)(4)式可得到定理的結(jié)論. 】

2.2 l=2的情形

表2 m為偶數(shù)時的重量分布

證明分以下4種情況來確定N(a,b)的值.

( i )若a=b=0,則由(5)式可得

由引理6可知該值出現(xiàn)的次數(shù)為

(iv)若a∈FqFp,b∈Fq或a=0,b∈F*q,則

顯然該值出現(xiàn)的次數(shù)為(q-p)q+q-1,即

pm(pm-p+1)-1.

由(4)式可得到碼的重量分布. 】

證明分以下3種情況來確定N(a,b)的值.

( i )若a=b=0,則

由引理6可知該值出現(xiàn)的次數(shù)為

由(4)式可定義

其對應(yīng)的重數(shù)分別為Aw1,Aw2,Aw3,根據(jù)MacWilliams方程[14]可得

解方程可得該碼的重量分布. 】

2.3 l=pm/2+1的情形

令m=2s(s為一個整數(shù)).與l=2的情形類似可知,若a?F*p則Ω3=0.若a∈F*p則由引理5有

表的重量分布

證明依據(jù)引理7,該定理的證明方法與定理2的類似,此處不再贅述. 】

3 結(jié)束語

根據(jù)文獻[6]的結(jié)論,文中構(gòu)造的線性碼可應(yīng)用于秘密共享方案.

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