Linear codes over Fq which are equivalent to LCD codes
Linear codes over Fq which are equivalent to LCD codes
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发表时间：
2017-03
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通讯作者：
C. Carlet;Sihem Mesnager;Chunming Tang;Yanfeng Qi
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作者：
C. Carlet;Sihem Mesnager;Chunming Tang;Yanfeng Qi
具有互补码的线性码（简称LCD）是指与其对偶的交是平凡的线性码。当它们是二进制时，它们在保护实现免受侧通道攻击和故障注入攻击方面发挥着重要作用。特征2中的非二进制LCD码可以通过扩展转换为二进制LCD码。本文介绍了由任意线性码构造LCD码的一般方法。进一步，我们证明了$\mathbb F_{q}（q>3）$上的任何线性码都等价于欧几里德LCD码，$\mathbb F_{q^2}（q>2）$上的任何线性码都等价于厄米特LCD码.因此，如果存在$\mathbb F_q$上的$[n，k，d]$-线性码，则存在$\mathbb F_q $上的$[n，k，d]$-线性欧几里德LCD码，其中$q>3$;如果存在$\mathbb F_{q^2}$上的$[n，k，d]$-线性码，则存在$\mathbb F_{q^2}$上的$[n，k，d]$-线性厄米特LCD码，其中$q>2$。因此，当$q>3$（相应地，q>2$）$q$-ary Euclidean（resp. $q ^2 $-ary Hermitian）LCD码具有与$q$-ary线性码相同的渐近界（分别为：$q^2$-ary linear codes）。最后，我们提出了一种通过扩展线性码构造LCD码的方法。
Linear codes with complementary duals (abbreviated LCD) are linear codes whose intersection with their dual are trivial. When they are binary, they play an important role in armoring implementations against side-channel attacks and fault injection attacks. Non-binary LCD codes in characteristic 2 can be transformed into binary LCD codes by expansion. In this paper, we introduce a general construction of LCD codes from any linear codes. Further, we show that any linear code over $\mathbb F_{q} (q>3)$ is equivalent to an Euclidean LCD code and any linear code over $\mathbb F_{q^2} (q>2)$ is equivalent to a Hermitian LCD code. Consequently an $[n,k,d]$-linear Euclidean LCD code over $\mathbb F_q$ with $q>3$ exists if there is an $[n,k,d]$-linear code over $\mathbb F_q$ and an $[n,k,d]$-linear Hermitian LCD code over $\mathbb F_{q^2}$ with $q>2$ exists if there is an $[n,k,d]$-linear code over $\mathbb F_{q^2}$. Hence, when $q>3$ (resp.$q>2$) $q$-ary Euclidean (resp. $q^2$-ary Hermitian) LCD codes possess the same asymptotical bound as $q$-ary linear codes (resp. $q^2$-ary linear codes). Finally, we present an approach of constructing LCD codes by extending linear codes.