Construction of Multicyclic Codes of Arbitrary Dimension $r$ via Idempotents: A Unified Combinatorial-Algebraic Approach

TL;DR

This paper introduces a unified algebraic-combinatorial method for constructing multicyclic codes of any dimension over finite fields, using tensor product idempotents and cyclotomic orbits, with applications to optimal codes.

Contribution

It develops a novel approach combining tensor product idempotents and cyclotomic orbits to construct multicyclic codes of arbitrary dimension, unifying algebraic and combinatorial perspectives.

Findings

Establishes a direct equivalence between combinatorial and algebraic descriptions of multicyclic codes.

Provides a natural polynomial basis for these codes.

Offers an optimal product bound that generalizes BCH and Reed-Solomon bounds.

Abstract

We propose a unified method to construct multicyclic codes of arbitrary dimension $r$ over $\mathbb{F}_q$. The approach relies on $r$-dimensional primitive idempotents defined as tensor products of univariate ones, combined with multidimensional cyclotomic orbits. This establishes a direct equivalence between combinatorial and algebraic descriptions, yields a natural polynomial basis, and provides an optimal product bound generalizing BCH and Reed-Solomon bounds. An efficient constructive algorithm is presented and illustrated by optimal 3-dimensional codes.