Schur Products of Constacyclic Codes via the Constacyclic Discrete Fourier Transform

TL;DR

This paper explores the Schur product of constacyclic codes using a specialized Fourier transform, extending existing methods and analyzing their properties through combinatorial structures.

Contribution

It introduces a constacyclic discrete Fourier transform, generalizes methods for code multiplication, and links Schur product properties to additive combinatorics.

Findings

Characterized properties of the constacyclic DFT.

Extended methods for computing the Schur product of constacyclic codes.

Derived properties of the Schur product dimension from combinatorial structures.

Abstract

This paper investigates the Schur product of constacyclic codes via the constacyclic discrete Fourier transform (DFT). We first characterize key properties of the constacyclic DFT, highlighting its differences from the ordinary DFT. We then extend the concept of degenerate cyclic codes to constacyclic codes possessing a nontrivial pattern polynomial, thereby facilitating the analysis of their dimension sequences. Building on these tools, we generalize two established methods for computing the square of cyclic codes to compute the Schur product of arbitrary constacyclic codes. Finally, exploiting the inherent combinatorial structure, we derive properties of the Schur product dimension directly from additive combinatorics.