2024/03/31 by Awadhesh Kumar Shukla, Shukla, Awadhesh Kumar, Sachin Pathak +7
Computer Science · Engineering · #Coding theory and cryptography #graph theory and CDMA systems #Quantum-Dot Cellular Automata
paper · pdf · doi:10.48550/arxiv.2404.00613
Let \mathbb Fq be a finite field, where q is an odd prime power. Let R=\mathbbFq+u\mathbbFq+v\mathbbFq+uv\mathbb Fq with u2=u,v2=v,uv=vu. In this paper, we study the algebraic structure of (θ, Θ)-cyclic codes of block length (r,s ) over \mathbbFqR. Specifically, we analyze the structure of these codes as left R[x:Θ]-submodules of \mathfrakRr,s = \frac\mathbbFq[x:θ]⟨ xr-1⟩ × (R[x:Θ])/(⟨ xs-1⟩). Our investigation involves determining generator polynomials and minimal generating sets for this family of codes. Further, we discuss the algebraic structure of separable codes. A relationship between the generator polynomials of (θ, Θ)-cyclic codes over \mathbb FqR and their duals is established. Moreover, we calculate the generator polynomials of dual of (θ, Θ)-cyclic codes. As an application of our study, we provide a construction of quantum error-correcting codes (QECCs) from (θ, Θ)-cyclic codes of block length (r,s) over \mathbbFqR. We support our theoretical results with illustrative examples.