Two New Families of Two-Weight Codes
Two New Families of Two-Weight Codes
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DOI：
10.1109/tit.2017.2742499
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发表时间：
2016-12
影响因子：
2.5
通讯作者：
M. Shi;Y. Guan;P. Solé
中科院分区：
文献类型：
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作者：
M. Shi;Y. Guan;P. Solé
我们构造了两个尺寸跟踪代码$ 2M $的新无限家族，ring $ \ mathbb {f} _ {p}+u \ mathbb {f} _ {p} $，带有$ u^{2} = u $，当$ p $是一个奇怪的素数时。它们具有阿贝尔法规的代数结构。他们的Lee重量分布是通过使用高斯总和计算的。通过灰色映射，我们获得了两个无限族的线性$ p $ -ary的代码$（p^{m} -1）-1）^{2} $和$ 2（p^{m} -1）^{2} {2} $。当$ m $甚至单一时，第一个家庭提供了五次重量代码。当$ m $是奇数和$ p \ equiv 3 \ pmod {4} $时，第一家族产生$ p $ - y -ary两重量代码，通过应用Griesmer绑定，这证明是最佳的。第二个家庭由Griesmer Bound，每当$ p = 3 $和$ m \ ge 3 $或$ p \ ge 5 $和$ m \ ge 4 $时。给出了秘密共享计划的申请。
We construct two new infinite families of trace codes of dimension $2m$ , over the ring $\mathbb {F}_{p}+u\mathbb {F}_{p}$ , with $u^{2}=u$ , when $p$ is an odd prime. They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using Gauss sums. By Gray mapping, we obtain two infinite families of linear $p$ -ary codes of respective lengths $(p^{m}-1)^{2}$ and $2(p^{m}-1)^{2}$ . When $m$ is singly even, the first family gives five-weight codes. When $m$ is odd and $p\equiv 3 \pmod {4}$ , the first family yields $p$ -ary two-weight codes, which are shown to be optimal by application of the Griesmer bound. The second family consists of two-weight codes that are shown to be optimal, by the Griesmer bound, whenever $p=3$ and $m \ge 3$ , or $p\ge 5$ and $m\ge 4$ . Applications to secret sharing schemes are given.