2024/12/26 by Akanksha, Sarma, Ritumoni
#13M05 #94B05 #94B15 #94B99 #FOS: Computer and information sciences #Information Theory (cs.IT)
paper · doi:10.48550/arxiv.2412.19126
In this article, for the finite field \mathbbFq, we show that the \mathbbFq-algebra \mathbbFq[x]/⟨ f(x) ⟩ is isomorphic to the product ring \mathbbFq°f(x) if and only if f(x) splits over \mathbbFq into distinct factors. We generalize this result to the quotient of the polynomial algebra \mathbbFq[x1, x2,…, xk] by the ideal ⟨ f1(x1), f2(x2),…, fk(xk)⟩. On the other hand, we establish that every finite-dimensional \mathbbFq-algebra S has an orthogonal basis of idempotents with their sum equal to 1S if and only if S≅\mathbbFql as \mathbbFq-algebras, where l=dim_\mathbbFq S. Instead of studying polycyclic codes over \mathbbFq-algebras \mathbbFq[x1, x2,…, xk]/⟨ f1(x1), f2(x2),…, fk(xk)⟩ where fi(xi) splits into distinct linear factors over \mathbbFq, which is a subclass of \mathbbFql, we study polycyclic codes over \mathbbFql and obtain their unique decomposition into polycyclic codes over \mathbbFq for every such orthogonal basis of \mathbbFql. We refer to it as an \mathbbFq-decomposition. An \mathbbFq-decomposition enables us to use results of polycyclic codes over \mathbbFq to study polycyclic codes over \mathbbFql; for instance, we show that the annihilator dual of a polycyclic code over \mathbbFql is a polycyclic code over \mathbbFql. Furthermore, with the help of different Gray maps, we produce a good number of examples of MDS or almost-MDS or/and optimal codes; some of them are LCD over \mathbbFq. Finally, we study Gray maps from (\mathbbFql)n to \mathbbFqnl, and use it to construct quantum codes with the help of CSS construction.