On the structure of cyclic codes over M2 $$({\mathbb {F}}_{\it{p}}$$ ( F p + u $${\mathbb {F}}_{\it{p}})$$ F p )

Habibul Islam (), Om Prakash () and Dipak Kumar Bhunia ()
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Habibul Islam: Indian Institute of Technology Patna
Om Prakash: Indian Institute of Technology Patna
Dipak Kumar Bhunia: Indian Institute of Technology Patna

Indian Journal of Pure and Applied Mathematics, 2022, vol. 53, issue 1, 153-161

Abstract: Abstract In this article, for a prime p such that $$p\equiv 2$$ p ≡ 2 or $$3 \pmod {5}$$ 3 ( mod 5 ) , we identify cyclic codes of length N over $$R=M_{2}({\mathbb {F}}_{p}+u{\mathbb {F}}_{p})$$ R = M 2 ( F p + u F p ) , $$u^2=0$$ u 2 = 0 as right R-submodules of $$R[x]/\langle x^N-1\rangle $$ R [ x ] / ⟨ x N - 1 ⟩ . Also, we define an isometry from $$M_{2}({\mathbb {F}}_{p}+u{\mathbb {F}}_{p})$$ M 2 ( F p + u F p ) to $${\mathbb {F}}_{p^2}+u{\mathbb {F}}_{p^2}+v{\mathbb {F}}_{p^2}+uv{\mathbb {F}}_{p^2}$$ F p 2 + u F p 2 + v F p 2 + u v F p 2 , where $$u^2=v^2=0,uv=vu$$ u 2 = v 2 = 0 , u v = v u and determine the structure of cyclic codes, in particular self-dual cyclic codes of length N where $$\gcd (N,p)=1$$ gcd ( N , p ) = 1 . Moreover, several optimal and near to optimal codes are obtained as the Gray images of these codes over R.

Keywords: Cyclic code; Gray map; Self-dual code; Matrix ring; Optimal code; 94B05; 94B15; 94B35; 94B60 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s13226-021-00014-x

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