Codes over an infinite family of local rings of order $$2^{5^k}$$ with two Gray maps

Codes over a family of local Frobenius rings, Gray maps and self-dual codes

Quasi-cyclic codes as cyclic codes over a family of local rings

We study cyclic codes over commutative local Frobenius rings of order 16 and give their binary images under a Gray map which is a generalization of the Gray maps on the rings of order 4. We prove that the binary images of cyclic codes are quasi-cyclic codes of index 4 and give examples of cyclic codes of various lengths constructed from these techniques including new optimal quasi-cyclic codes.

In this work, we study constacyclic codes of odd length over finite commutative local Frobenius non-chain rings of order 16. 16. We give the structure of λ i \lambda _i -constacyclic codes over each local Frobenius non-chain ring via cyclic codes over these rings where λ i \lambda _i acts as a weight preserving unit in the corresponding ring. We also obtain binary optimal codes which are images of these constacyclic codes under a gray map.

Constacyclic codes are an important class of linear codes in coding theory. Many optimal linear codes are directly derived from constacyclic codes. In this paper, (1 − uv)-constacyclic codes over the local ring \(\mathbb{F}_p + u\mathbb{F}_p + v\mathbb{F}_p + uv\mathbb{F}_p \) are studied. It is proved that the image of a (1 − uv)-constacyclic code of length n over \(\mathbb{F}_p + u\mathbb{F}_p + v\mathbb{F}_p + uv\mathbb{F}_p \) under a Gray map is a distance invariant quasi-cyclic code of index p2 and length p3n over \(\mathbb{F}_p \). Several examples of optimal linear codes over \(\mathbb{F}_p \) from (1 − uv)-constacyclic codes over \(\mathbb{F}_p + u\mathbb{F}_p + v\mathbb{F}_p + uv\mathbb{F}_p \) are given.

A new class of optimal linear codes with flexible parameters

In this paper, we study cyclic codes and their duals over the local Frobenius non-chain ring $$R={\mathbb {F}}_2[u,v] / \langle u^2=v^2,uv \rangle $$, and we obtain optimal binary linear codes with respect to the homogeneous weight over R via a Gray map. Moreover, we characterize DNA codes as images of cyclic codes over R.

In this paper, (1+ϑ)-constacyclic codes of arbitrary length m over a non-chain finite local frobenious ring Z8 + ϑZ8 are introduced. A new Gray map is constructed from Z8 + ϑZ8 to Z88 and proved that the Z8 - Gray image of (1+ϑ)-constacyclic codes having prescribed length m over the ring Z8 +ϑZ8 is a cyclic code of length 8m over the ring Z8. Moreover, it has been obtained that the binary image of the (1+ϑ)-constacyclic code of length m over Z8 +ϑZ8 is a distance invariant binary quasi-cyclic code of length 32m with index 16.

In this paper, (1+ϑ)-constacyclic codes of arbitrary length m over a non-chain finite local frobenious ring Z 8 + ϑZ 8 are introduced. A new Gray map is constructed from Z 8 + ϑZ 8 to Z 8 8 and proved that the Z 8 - Gray image of (1+ϑ)-constacyclic codes having prescribed length m over the ring Z 8 +ϑZ 8 is a cyclic code of length 8m over the ring Z 8 . Moreover, it has been obtained that the binary image of the (1+ϑ)-constacyclic code of length m over Z 8 +ϑZ 8 is a distance invariant binary quasi-cyclic code of length 32m with index 16.

We study the algebraic structure of DNA codes constructed over the ring R=F_2 [u,v,w] ⟨u^2=v^2,uv=0,w^2=w⟩, which is a commutative local Frobenius non-chain ring. We define a gray map over R and generate DNA codes using the images of the gray map. We define reversible DNA codes and reversible complement DNA codes over the ring.

Weight preserving Gray maps are defined from any non-chain local ring of order 16 to the binary Hamming space. This is used to define the Lee weight for codes in this setting. MacWilliams relations for the weight enumerator with respect to the Lee weight are given. Self-dual codes are studied over these rings and they are used to study binary codes whose weight enumerators are held invariant by the action of the MacWilliams relations.

Two families of few-weight codes over a finite chain ring

We construct an infinite family of two-Lee-weight codes over the ring $\mathbb {F}_{2}+u\mathbb {F}_{2}$ . These codes are defined as trace codes. They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using character sums. By Gray mapping, we obtain an infinite family of abelian binary two-weight codes. They are shown to be optimal by application of the Griesmer bound. An application to secret sharing schemes is given.

Few-weight codes over a non-chain ring associated with simplicial complexes and their distance optimal Gray image

Recently, some infinite families of binary minimal and optimal linear codes were constructed from simplicial complexes by Hyun et al. Inspired by their work, we present two new constructions of codes over the ring F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> + uF <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> by employing simplicial complexes. When the simplicial complexes are all generated by a maximal element, we determine the Lee weight distributions of two classes of the codes over F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> + uF <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> . Our results show that the codes have few Lee weights. Via the Gray map, we obtain an infinite family of binary codes meeting the Griesmer bound and a class of binary distance optimal codes.