New DNA Codes from Cyclic Codes over Mixed Alphabets

Let R=F4+uF4,withu2=u and S=F4+uF4+vF4,withu2=u,v2=v,uv=vu=0. In this paper, we study F4RS-cyclic codes of block length (α,β,γ) and construct cyclic DNA codes from them. F4RS-cyclic codes can be viewed as S[x]-submodules of Fq[x]⟨xα−1⟩×R[x]⟨xβ−1⟩×S[x]⟨xγ−1⟩. We discuss their generator polynomials as well as the structure of separable codes. Using the structure of separable codes, we study cyclic DNA codes. By using Gray maps ψ1 from R to F42 and ψ2 from S to F43, we give a one-to-one correspondence between DNA codons of the alphabets {A,T,G,C}2,{A,T,G,C}3 and the elements of R,S, respectively. Then we discuss necessary and sufficient conditions of cyclic codes over F4, R, S and F4RS to be reversible and reverse-complement. As applications, we provide examples of new cyclic DNA codes constructed by our results.

Skew cyclic and skew constacyclic codes over the ring 𝔽p + u1𝔽p + ⋯ + u2m𝔽p

In this paper, we study skew cyclic and skew constacyclic codes over the ring [Formula: see text], where [Formula: see text] and [Formula: see text], for [Formula: see text] and [Formula: see text]. We have introduced some new Gray maps and determined the relation between codes using the [Formula: see text]-Gray images of skew constacyclic codes.

Double Circulant Self-Dual and LCD Codes Over ℤp2

We study double circulant LCD codes over [Formula: see text] for all odd primes [Formula: see text] and self-dual double circulant codes over [Formula: see text] for primes [Formula: see text]. We derive exact enumeration formulae, and asymptotic lower bounds on the minimum distance of the [Formula: see text]-ary images of these codes by the classical Gray maps.

A class of constacyclic codes over the ring Z4[u,v]/ and their Gray images

In this paper, we study (1 + 2u + 2v)-constacyclic and skew (1 + 2u + 2v)-constacyclic codes over the ring Z4 + uZ4 + vZ4 + uvZ4 where u2 = v2 = 0,uv = vu. We define some new Gray maps and show that the Gray images of (1 + 2u + 2v)-constacyclic and skew (1 + 2u + 2v)-constacyclic codes are cyclic, quasi-cyclic and permutation equivalent to quasi-cyclic codes over Z4. Further, we determine the structure of (1 + 2u + 2v)-constacyclic codes of odd length n.

Some Constacyclic Codes over ℤ4[u]/〈u2〉,, New Gray Maps, and New Quaternary Codes

In this paper, we study λ-constacyclic codes over the ring R = ℤ4 + uℤ4, where u2 = 0, for λ =1 + 3u and 3 + u. We introduce two new Gray maps from R to [Formula: see text] and show that the Gray images of λ-constacyclic codes over R are quasi-cyclic over ℤ4. Moreover, we present many examples of λ-constacyclic codes over R whose ℤ4-images have better parameters than the currently best-known linear codes over ℤ4.

A study of constacyclic codes over the ring ℤ4[u]/〈u2 − 3〉

In this paper, we study [Formula: see text]-constacyclic codes over the ring [Formula: see text], where [Formula: see text] for [Formula: see text] and [Formula: see text], respectively. We define some new Gray maps from [Formula: see text] to the copies of [Formula: see text]. It is shown that Gray images of [Formula: see text]-constacyclic codes over [Formula: see text] are cyclic, quasi-cyclic and permutation equivalent to quasi-cyclic codes over [Formula: see text]. Further, we extend and obtain cyclic codes, [Formula: see text]-constacyclic codes and permutation equivalent to quasi-cyclic codes over [Formula: see text], respectively, as Gray images of skew [Formula: see text]-constacyclic codes over [Formula: see text].

Some results on quadratic residue codes over the ring Fp + vFp + v2Fp + v3Fp

Quadratic residue (QR) codes and their extensions over the finite non-chain ring [Formula: see text] are studied, where [Formula: see text], [Formula: see text] is an odd prime and [Formula: see text]. A class of Gray maps preserving the self-duality of linear codes from [Formula: see text] to [Formula: see text] is given. Under a special Gray map, a self-dual code [Formula: see text] over [Formula: see text], a formally self-dual code [Formula: see text] over [Formula: see text] and a formally self-dual code [Formula: see text] over [Formula: see text] are obtained from extended QR codes.