Coding Theory using Linear Complexity of Finite Sequences

TL;DR

This paper introduces a new coding theory based on a metric derived from the linear complexity of finite sequences, providing bounds, constructions, and size calculations relevant for cryptography.

Contribution

It develops a novel metric on finite sequences, establishes a Singleton-like bound, and constructs optimal subspace codes for cryptographic applications.

Findings

Established a new metric based on linear complexity

Derived a Singleton-like bound for codes in this metric

Constructed codes that achieve the bound

Abstract

We define a metric on $\mathbb{F}_q^n$ using the linear complexity of finite sequences. We will then develop a coding theory for this metric. We will give a Singleton-like bound and we will give constructions of subspaces of $\mathbb{F}_q^n$ achieving this bound. We will compute the size of balls with respect to this metric. In other words we will count how many finite sequences have linear complexity bounded by some integer $r$. The paper is motivated in part by the desire to design new code based cryptographic systems.