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arXiv:1711.03307

Self-Dual Cyclic and Quantum Codes Over Z2^α x (Z2 + uZ2)^β

İsmail Aydoğdu, Taher Abualrub

Published Nov 9, 2017• 1 citation
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Abstract

In this paper we introduce self-dual cyclic and quantum codes over Z2^α x (Z2 + uZ2)^β. We determine the conditions for any Z2Z2[u]-cyclic code to be self-dual, that is, C = C^{\perp}. Since the binary image of a self-orthogonal Z2Z2[u]-linear code is also a self-orthogonal binary linear code, we introduce quantum codes over Z2^α x (Z2 + uZ2)^β. Finally, we present some examples of self-dual cyclic and quantum codes that have good parameters.

DOI
10.48550/arxiv.1711.03307

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In this paper we introduce self-dual cyclic and quantum codes over Z2^α x (Z2 + uZ2)^β. We determine the conditions for any Z2Z2[u]-cyclic code to be self-dual, that is, C = C^{\perp}. Since the binary image of a self-orthogonal Z2Z2[u]-linear code is also a self-orthogonal binary linear code, we introduce quantum codes over Z2^α x (Z2 + uZ2)^β. Finally, we present some examples of self-dual cyclic and quantum codes that have good parameters.
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We present new constructions of binary quantum codes from quaternary linear Hermitian self-dual codes. Our main ingredients for these constructions are nearly self-orthogonal cyclic or duadic codes over F_4. An infinite family of 000-dimensional binary quantum codes is provided. We give minimum distance lower bounds for our quantum codes in terms of the minimum distance of their ingredient linear codes. We also present new results on the minimum distance of linear cyclic codes using their fixed subcodes. Finally, we list many new record-breaking quantum codes obtained from our constructions.
In this paper, we study the structural properties of ( α + u 1 β + u 2 γ + u 1 u 2 δ ) -constacyclic codes over R = F q [ u 1 , u 2 ] / ⟨ u 1 2 − u 1 , u 2 2 − u 2 , u 1 u 2 − u 2 u 1 ⟩ where q = p m for odd prime p and m ≥ 1 . We derive the generators of constacyclic and dual constacyclic codes. We have shown that Gray image of a constacyclic code of length n is a quasi constacyclic code of length 4 n . Also we have classified all possible self dual linear codes over this ring R . We have given the applications by computing non-binary quantum codes over this ring R .
We give a polynomial representation for additive cyclic codes over $ \mathbb{F}_{p^2} $. This representation will be applied to uniquely present each additive cyclic code by at most two generator polynomials. We determine the generator polynomials of all different additive cyclic codes. A minimum distance lower bound for additive cyclic codes will also be provided using linear cyclic codes over $ \mathbb{F}_p $. We classify all the symplectic self-dual, self-orthogonal, and nearly self-orthogonal additive cyclic codes over $ \mathbb{F}_{p^2} $. Finally, we present ten record-breaking binary quantum codes after applying a quantum construction to self-orthogonal and nearly self-orthogonal additive cyclic codes over $ \mathbb{F}_{4} $.