Squares of Random Linear Codes

TL;DR

This paper investigates when the square of a random linear code typically spans the entire space, showing that for certain dimensions and lengths, it does so with high probability, using combinatorial and algebraic methods.

Contribution

It provides a probabilistic analysis demonstrating that the square of a linear code usually fills the whole space under specific size constraints, with exponential convergence speed.

Findings

Squares of random codes fill the entire space with high probability.

Convergence to full space is exponential when code dimension and length satisfy certain conditions.

Uses combinatorial and algebraic tools to analyze quadratic forms and their zeros.

Abstract

Given a linear code $C$, one can define the $d$-th power of $C$ as the span of all componentwise products of $d$ elements of $C$. A power of $C$ may quickly fill the whole space. Our purpose is to answer the following question: does the square of a code "typically" fill the whole space? We give a positive answer, for codes of dimension $k$ and length roughly $\frac{1}{2}k^2$ or smaller. Moreover, the convergence speed is exponential if the difference $k(k+1)/2-n$ is at least linear in $k$. The proof uses random coding and combinatorial arguments, together with algebraic tools involving the precise computation of the number of quadratic forms of a given rank, and the number of their zeros.