Self-Dual Near MDS Codes from Elliptic Curves
Self-Dual Near MDS Codes from Elliptic Curves
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DOI：
10.1109/tit.2018.2880913
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发表时间：
2019-04
影响因子：
2.5
通讯作者：
Lingfei Jin;Haibin Kan
中科院分区：
文献类型：
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作者：
Lingfei Jin;Haibin Kan
近年来，自对偶MDS码由于其理论意义和实际应用价值而引起了广泛的关注。与自对偶MDS码类似，自对偶近MDS码也具有良好的结构。从理论和实际的角度来看，研究自对偶NMDS码是很自然的。尽管文献中对NMDS码进行了大量的研究，但对于自对偶NMDS码却知之甚少。构造自对偶NMDS码比构造自对偶MDS码更具挑战性。关于自对偶NMDS码的唯一构造工作表明，对于奇素数幂q，存在长度为q-1的q元自对偶NMDS码，对于q为197的小素数q，存在长度为16的q元自对偶NMDS码。本文利用椭圆曲线的性质构造了自对偶NMDS码。事实证明，只要2美元|q$和$n$是偶数，其中$4\len\leq +\lfloor 2\sqrt {q}\rfloor-2 $，可以构造出长度为$n$的自对偶NMDS码。此外，对于奇素数幂q，存在环q上长度为n的自对偶NMDS码，如果q\ge 4^{n+3}\times（n+3）^{2}$。
In recent years, self-dual MDS codes have attracted a lot of attention due to theoretical interest and practical importance. Similar to self-dual MDS codes, self-dual near MDS (NMDS for short) codes have nice structures as well. From both theoretical and practical points of view, it is natural to study self-dual NMDS codes. Although there has been lots of work on NMDS codes in literature, little is known for self-dual NMDS codes. It seems more challenging to construct self-dual NMDS codes than self-dual MDS codes. The only work on construction of self-dual NMDS codes shows existence of $q$ -ary self-dual NMDS codes of length $q-1$ for odd prime power $q$ or length up to 16 for some small primes $q$ with $q\le 197$ . In this paper, we make use of properties of elliptic curves to construct self-dual NMDS codes. It turns out that, as long as $2|q$ and $n$ is even with $4\le n\le q+\lfloor 2\sqrt {q}\rfloor -2$ , one can construct a self-dual NMDS code of length $n$ over $ {\mathbb {F}}_{q}$ . Furthermore, for odd prime power $q$ , there exists a self-dual NMDS code of length $n$ over $ {\mathbb {F}}_{q}$ if $q\ge 4^{n+3}\times (n+3)^{2}$ .