Reversible DNA codes from skew cyclic codes over a ring of order 256

We introduce skew cyclic codes over the finite ring $\R$, where $u^{2}=0,v^{2}=v,w^{2}=w,uv=vu,uw=wu,vw=wv$ and use them to construct reversible DNA codes. The 4-mers are matched with the elements of this ring. The reversibility problem for DNA 4-bases is solved and some examples are provided

MacDonald codes over the ring F3+ vF3

The binary MacDonald codes were introduced in [9] and q− ary version (q≥ 2) MacDonald code over the finite field Fq was studied in [10]. In [5], CJ Colbourn and M. Gupta obtained two families of MacDonald codes over the ring Z4 from Z4-simplex codes of types α and β, Sα k and S β k . They studied some fundamental properties of the codes. In [1], it was shown that the results of [5] concerning the codes over the ring Z4 are valid for the ring F2+ uF2 where u2= 0 and F2 is a field of two elements. In [2], the Mac-Donald codes over the ring F2+ uF2+ u2F2 were constructed, where u3= 0 and F2={0, 1} by using simplex codes over the ring F2+ uF2+ u2F2. Their properties were described. In [6], the MacDonald codes over F2+ vF2 wer

On Linear Codes Over a Non-Chain Ring

In this paper, we study skew cyclic and quasi cyclic codes over the ring S = F2 +uF2 + vF2 where u2 = u, v2 = v, uv = vu = 0.We investigate the structural properties of them. Using a Gray map on S we obtain the MacWilliams identities for codes over S. The relationships between Symmetrized, Lee and Hamming weight enumerator are determined

On (1 − u m )-Cyclic Codes over

Abstract A new Gray map between codes over F 2 + uF 2 + u 2 F 2 + u 3 F 2 + .... + u m F 2 and codes over F 2 is defined. It is proved that the Gray image of a linear (1 − u m )-cyclic code over of length n is a binary distance invariant quasi-cylic code of index 2 m−1 and length 2 m n. It is also proved that if n is odd, then every code of length 2 m n over F 2 which is the Gray image of a linear cyclic code of length n over F 2 + uF 2 + u 2 F 2 + u 3 F 2 + .... + u m F 2 is equivalent to a quasi-cyclic code of index 2 m−1 . Mathematics Subject Classification: 94B15, 94B6

On cyclic DNA codes over the rings Z4 + wZ4 and Z4 + wZ4 + vZ4 + wvZ4

The structures of cyclic DNA codes of odd length over the finite rings R = Z4 + wZ4, w^2 = 2 and S = Z4 + wZ4 + vZ4 + wvZ4; w^2 = 2; v^2 =v; wv = vw are studied. The links between the elements of the rings R, S and 16 and 256 codons are established, respectively. The cyclic codes of odd length over the finite ring R satisfy reverse complement constraint and the cyclic codes of odd length over the finite ring S satisfy reverse constraint and reverse complement constraint are studied. The binary images of the cyclic DNA codes over the finite rings R and S are determined. Moreover, a family of DNA skew cyclic codes over R is constructed, its property of being reverse complement is studied

Simplex Codes of Type γ over F3+ vF3

In this paper, it is constructed simplex linear codes over the ring F3+ vF3 of type γ, where v 2= 1 and F3={0, 1, 2} and obtained the minimum Hamming, Lee and Bachoc weights of this codes

A new matrix form to generate all 3 x 3 involutory MDS matrices over F-2(m)

© 2019 Elsevier B.V. In this paper, we propose a new matrix form to generate all 3×3 involutory and MDS matrices over F 2 m and prove that the number of all 3×3 involutory and MDS matrices over F 2 m is (2 m −1) 2 ⋅(2 m −2)⋅(2 m −4), where m>2. Moreover, we give 3×3 involutory and MDS matrices over F 2 3 , F 2 4 and F 2 8 defined by the irreducible polynomials x 3 +x+1, x 4 +x+1 and x 8 +x 7 +x 6 +x+1, respectively, by considering the minimum XOR count, which is a metric used in the estimation of hardware implementation cost. Finally, we provide the maximum number of 1s in 3×3 involutory MDS matrices.status: publishe