On self-dual and LCD double circulant and double negacirculant codes over Fq+uFq$\mathbb {F}_{q}+u\mathbb {F}_{q}$

Self-dual and LCD double circulant and double negacirculant codes over $${\mathbb {F}}_q+u{\mathbb {F}}_q+v{\mathbb {F}}_q$$

“…Proof The proof depends on the main fact φ i (C ⊥ ) = (φ i (C)) ⊥ , which can be verified by using the similar procedure of [ [21], Theorem 2.2].…”

“…(see [21,28]) where a n = 8(1 + q (n−1)/2 ) 2 for self-dual and q 2n+1 , for LCD codes. Therefore, by Lemma 3 and Lemma 4 we conclude that in the family, there exist codes of length 2n over R whose images under φ i have Hamming distance ≥ d n .…”

“…Meanwhile, self-dual negacirculant codes over finite fields were studied in [1]. To generalize the concept over finite rings, recently, Shi et al [21] considered the finite commutative semi-local non-chain ring F q + uF q , u 2 = u and studied double circulant self-dual or LCD codes. They first enumerated these codes for self-dual and LCD codes, respectively, and later obtained distance bounds on them.…”

“…Further, for some related studies on these topics, interested readers can see [20,24,26,27,29]. Therefore, because of available works on non-chain rings [21,28], it is logical to investigate these codes over other semi-local non-chain rings. Motivated by the above studies, here we consider the finite commutative semi-local non-chain ring R = F q + uF q + vF q , where u 2 = u, v 2 = v, uv = vu = 0.…”

On Self‐dual and LCD Double Circulant Codes over a Non‐chain Ring*

“…[7] and Ref. [8], respectively. Quasicyclic codes over fields have been proved to contain infinite families of good long LCD codes Ref.…”