Primitive Rateless Codes

TL;DR

This paper introduces primitive rateless (PR) codes, which are generated using primitive polynomials over GF(2), allowing for limitless coded symbols and comparable error correction performance to BCH codes, with applications in flexible data transmission.

Contribution

The paper proposes a novel class of primitive rateless codes characterized by primitive polynomials, providing a new approach to rate-compatible coding with theoretical analysis and practical performance evaluation.

Findings

PR codes can be represented as subsequences of m-sequences.

PR codes' Hamming weight distribution approximates a truncated binomial distribution.

PR codes achieve block error rates similar to BCH codes at various SNRs.

Abstract

In this paper, we propose primitive rateless (PR) codes. A PR code is characterized by the message length and a primitive polynomial over $\mathbf{GF}(2)$, which can generate a potentially limitless number of coded symbols. We show that codewords of a PR code truncated at any arbitrary length can be represented as subsequences of a maximum-length sequence ($m$-sequence). We characterize the Hamming weight distribution of PR codes and their duals and show that for a properly chosen primitive polynomial, the Hamming weight distribution of the PR code can be well approximated by the truncated binomial distribution. We further find a lower bound on the minimum Hamming weight of PR codes and show that there always exists a PR code that can meet this bound for any desired codeword length. We provide a list of primitive polynomials for message lengths up to $40$ and show that the respective PR codes closely meet the Gilbert-Varshamov bound at various rates. Simulation results show that PR codes can achieve similar block error rates as their BCH counterparts at various signal-to-noise ratios (SNRs) and code rates. PR codes are rate-compatible and can generate as many coded symbols as required; thus, demonstrating a truly rateless performance.