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

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.

Theorems & Definitions (14)

• Definition 3.1: Primitive $r$-dim idempotents
• Proposition 3.2
• proof
• Theorem 3.3: Spectral decomposition, blahut2003
• proof
• Definition 3.4: Orbit
• Definition 3.5: Generating idempotent
• Theorem 3.6: Equivalence
• proof
• Theorem 3.7: Polynomial basis