Typical sumsets of linear codes

TL;DR

This paper characterizes the asymptotic size of typical sumsets of linear codes over finite fields, revealing a threshold rate that determines whether the sumset size is quadratic or linear in the code size, with applications to multi-user communication.

Contribution

It provides a complete characterization of the asymptotic size of typical sumsets for linear codes and nested codes, identifying a rate threshold that influences sumset growth.

Findings

For rates below threshold, sumset size is roughly the square of code size.

For rates above threshold, sumset size is roughly the product of code size and a constant.

The threshold depends only on the alphabet size and lies within a specific range.

Abstract

Given two identical linear codes $\mathcal C$ over $\mathbb F_q$ of length $n$, we independently pick one codeword from each codebook uniformly at random. A $\textit{sumset}$ is formed by adding these two codewords entry-wise as integer vectors and a sumset is called $\textit{typical}$, if the sum falls inside this set with high probability. We ask the question: how large is the typical sumset for most codes? In this paper we characterize the asymptotic size of such typical sumset. We show that when the rate $R$ of the linear code is below a certain threshold $D$, the typical sumset size is roughly $|\mathcal C|^2=2^{2nR}$ for most codes while when $R$ is above this threshold, most codes have a typical sumset whose size is roughly $|\mathcal C|\cdot 2^{nD}=2^{n(R+D)}$ due to the linear structure of the codes. The threshold $D$ depends solely on the alphabet size $q$ and takes value in $[1/2, \log \sqrt{e})$. More generally, we completely characterize the asymptotic size of typical sumsets of two nested linear codes $\mathcal C_1, \mathcal C_2$ with different rates. As an application of the result, we study the communication problem where the integer sum of two codewords is to be decoded through a general two-user multiple-access channel.