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$${\pmb {{\mathbb {Z}}}}_p{\pmb {{\mathbb {Z}}}}_p[v]$$ Z p Z p [ v ] -additive cyclic codes are asymptotically good
Journal of Applied Mathematics and Computing ( IF 2.4 ) Pub Date : 2020-11-24 , DOI: 10.1007/s12190-020-01466-w
Xiaotong Hou , Jian Gao

We construct a class of \({\mathbb {Z}}_p{\mathbb {Z}}_p[v]\)-additive cyclic codes, where p is a prime number and \(v^2=v\). We determine the asymptotic properties of the relative minimum distance and rate of this class of codes. We prove that, for any positive real number \(0<\delta <1\) such that the p-ary entropy at \(\frac{k+l}{2}\delta \) is less than \(\frac{1}{2}\), the relative minimum distance of the random code is convergent to \(\delta \) and the rate of the random code is convergent to \(\frac{1}{k+l}\), where pkl are pairwise coprime positive integers.



中文翻译:

$$ {\ pmb {{\ mathbb {Z}}}} _ p {\ pmb {{\ mathbb {Z}}}} _ p [v] $$ Z p Z p [v]-附加循环代码在渐近性上很好

我们构造一类\({\ mathbb {Z}} _ p {\ mathbb {Z}} _ p [v] \)可加循环码,其中p是素数,\(v ^ 2 = v \)。我们确定此类代码的相对最小距离和速率的渐近性质。我们证明,对于任何正实数\(0 <\ delta <1 \),使得\(\ frac {k + 1} {2} \ delta \)p-熵小于\(\ frac {1} {2} \),随机码的相对最小距离收敛于\(\ delta \),随机码的速率收敛于\(\ frac {1} {k + 1} \)其中p,  k,  l 是成对的互质数正整数。

更新日期:2020-11-25
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