Properties of Constacyclic Codes Under the Schur Product

TL;DR

This paper investigates how constacyclic codes evolve under the Schur product, providing criteria for invariance and growth, which has implications for cryptography and coding theory.

Contribution

It characterizes the growth and invariance properties of constacyclic codes under the Schur product, a novel analysis in coding theory.

Findings

Identifies conditions for invariance of code powers under Schur product

Provides criteria for the growth of code dimensions

Enhances understanding of code structure in cryptography

Abstract

For a subspace $W$ of a vector space $V$ of dimension $n$, the Schur-product space $W^{\langle k\rangle}$ for $k \in \mathbb{N}$ is defined to be the span of all vectors formed by the component-wise multiplication of $k$ vectors in $W$. It is well known that repeated applications of the Schur product to the subspace $W$ creates subspaces $W, W^{\langle 2 \rangle}, W^{\langle 3 \rangle}, \ldots$ whose dimensions are monotonically non-decreasing. However, quantifying the structure and growth of such spaces remains an important open problem with applications to cryptography and coding theory. This paper characterizes how increasing powers of constacyclic codes grow under the Schur product and gives necessary and sufficient criteria for when powers of the code and or dimension of the code are invariant under the Schur product.