Linear codes over Fq which are equivalent to LCD codes

Abstract

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.

arXiv admin note: text overlap with arXiv:1702.08033

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Cited by in corpus (3)

• New characterization and parametrization of LCD Codes
• Explicit MDS Codes with Complementary Duals
• Binary LCD Codes from ${\mathbb{Z}}_2{\mathbb{Z}}_2[u]$