On the Hamming Weight Distribution of Subsequences of Pseudorandom Sequences

TL;DR

This paper analyzes the average Hamming weight distribution of subsequences of m-sequences, providing bounds and demonstrating their potential in constructing effective rateless codes.

Contribution

It characterizes the average Hamming weight distribution of subsequences of m-sequences and establishes bounds on their minimum Hamming weight, linking to code performance.

Findings

Average Hamming weight distribution characterized using MacWilliams identity

Lower bound on minimum Hamming weight of subsequences established

Proper primitive polynomials enable subsequences to form effective rateless codes

Abstract

In this paper, we characterize the average Hamming weight distribution of subsequences of maximum-length sequences ($m$-sequences). In particular, we consider all possible $m$-sequences of dimension $k$ and find the average number of subsequences of length $n$ that have a Hamming weight $t$. To do so, we first characterize the Hamming weight distribution of the average dual code and use the MacWilliams identity to find the average Hamming weight distribution of subsequences of $m$-sequences. We further find a lower bound on the minimum Hamming weight of the subsequences and show that there always exists a primitive polynomial to generate an $m$-sequence to meet this bound. We show via simulations that when a proper primitive polynomial is chosen, subsequences of the $m$-sequence can form a good rateless code that can meet the normal approximation benchmark.