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International Journal of Statistics and Applied Mathematics

Quantum codes derived from constacyclic codes over Z3 [u,v]/

Rekha, Monika Sangwan

Published Aug 1, 2025• 0 citations
Publisher

Abstract

The structural properties and construction methodologies of quantum codes (QCs) defined over the finite field Z_3, the finite field with three elements, are examined in this work. The primary focus is on utilizing constacyclic codes within a specific algebraic structure the finite commutative non-chain ring R=Z3+uZ3+vZ3+uvZ3, where the indeterminates u and v satisfy the relations u2=1, v2=1, and uv=vu. We provide a set of idempotent generators of the ring and define linear codes and calculated some self-inverse units. The relationship between R and Z34 is established using a gray map. Constacyclic (CC) codes are broken down into cyclic (C) codes and negacyclic (N) codes in order to determine the parameters of QCs over Z_3. Several QCs of arbitrary lengths are developed as an application. In this paper, we calculated those various parameters of QCs which are better than existing parameters of QCs. This paper contains only those units which have self-inverse.

DOI
10.22271/maths.2025.v10.i8c.2152

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The structural properties and construction methodologies of quantum codes (QCs) defined over the finite field Z_3, the finite field with three elements, are examined in this work. The primary focus is on utilizing constacyclic codes within a specific algebraic structure the finite commutative non-chain ring R=Z3+uZ3+vZ3+uvZ3, where the indeterminates u and v satisfy the relations u2=1, v2=1, and uv=vu. We provide a set of idempotent generators of the ring and define linear codes and calculated some self-inverse units. The relationship between R and Z34 is established using a gray map. Constacyclic (CC) codes are broken down into cyclic (C) codes and negacyclic (N) codes in order to determine the parameters of QCs over Z_3. Several QCs of arbitrary lengths are developed as an application. In this paper, we calculated those various parameters of QCs which are better than existing parameters of QCs. This paper contains only those units which have self-inverse.
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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 .