Cyclic codes over a non-chain ring R_e,q and their application to LCD codes
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Let 𝔽_q be a finite field of order q, a prime power integer such that q=et+1 where t≥ 1,e≥ 2 are integers. In this paper, we study cyclic codes of length n over a non-chain ring R_e,q=𝔽_q[u]/⟨ u^e-1⟩. We define a Gray map φ and obtain many maximum-distance-separable (MDS) and optimal 𝔽_q-linear codes from the Gray images of cyclic codes. Under certain conditions we determine linear complementary dual (LCD) codes of length n when (n,q)≠ 1 and (n,q)= 1, respectively. It is proved that a cyclic code 𝒞 of length n is an LCD code if and only if its Gray image φ(𝒞) is an LCD code of length 4n over 𝔽_q. Among others, we present the conditions for existence of free and non-free LCD codes. Moreover, we obtain many optimal LCD codes as the Gray images of non-free LCD codes over R_e,q.
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