Primitive Rateless Codes

Abstract

In this paper, we propose primitive rateless (PR) codes. A PR code is characterized by the message length and a primitive polynomial over 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.

Enhanced Distributed Storage System Using Lower Triangular Matrix-Luby Transform Codes

“…If the decoder can recover the original data from the minimum combination of possible encoding symbols, then LT codes are near optimal [8] with respect to any erasure channel conditions. The encoding and decoding times in LT codes are closely very efficient as a function of the data length [9]. Therefore, LT codes are near optimal for every erasure channel and they are very efficient as the data length grows [10].…”

Linear-Equation Ordered-Statistics Decoding

“…However, the ML decoding of LDPC codes has a high complexity [5]. A recently proposed primitive rateless (PR) code [11] with its inherent rateless property is also shown to perform very close to the finite length bounds; however, its decoding complexity is the major drawback. Furthermore, short BCH codes have also gained interest from the research community recently [5], [7], [12], [13], as it outperforms other existing short channel codes in terms of error correction capability.…”