Reliable channel regions for good binary codes transmitted over parallel channels

Research output: Contribution to journalArticlepeer-review

41 Scopus citations


We study the average error probability performance of binary linear code ensembles when each codeword is divided into J subcodewords with each being transmitted over one of J parallel channels. This model is widely accepted for a number of important practical channels and signaling schemes including block-fading channels, incremental redundancy retransmission schemes, and multicarrier communication techniques for frequency-selective channels. Our focus is on ensembles of good codes whose performance in a single channel model is characterized by a threshold behavior, e.g., turbo and low-density parity-check (LDPC) codes. For a given good code ensemble, we investigate reliable channel regions which ensure reliable communications over parallel channels under maximum-likelihood (ML) decoding. To construct reliable regions, we study a modifed 1961 Gallager bound for parallel channels. By allowing codeword bits to be randomly assigned to each component channel, the average parallel-channel Gallager bound is simplified to be a function of code weight enumerators and channel assignment rates. Special cases of this bound, average union-Bhattacharyya (UB), Shulman-Feder (SF), simplified-sphere (SS), and modified Shulman-Feder (MSF) parallel-channel bounds, allow for describing reliable channel regions using simple functions of channel and code spectrum parameters. Parameters describing the channel are the average parallel-channel Bhattacharyya noise parameter, the average channel mutual information, and parallel Gaussian channel signal-to-noise ratios (SNRs). Code parameters include the union-Bhattacharyya noise threshold and the weight spectrum distance to the random binary code ensemble. Reliable channel regions of repeat-accumulate (RA) codes for parallel binary erasure channels (BECs) and of turbo codes for parallel additive white Gaussian noise (AWGN) channels are numerically computed and compared with simulation results based on iterative decoding. In addition, an example transmission over a block-fading Gaussian channel is considered.

Original languageEnglish (US)
Pages (from-to)1405-1424
Number of pages20
JournalIEEE Transactions on Information Theory
Issue number4
StatePublished - Apr 2006

All Science Journal Classification (ASJC) codes

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences


  • Block-fading channel
  • Coding theorems
  • Hybrid automatic repeat request (HARQ)
  • Maximum-likelihood (ML) decoding
  • Parallel channels
  • Repeat- accumulate (RA) codes
  • Turbo codes


Dive into the research topics of 'Reliable channel regions for good binary codes transmitted over parallel channels'. Together they form a unique fingerprint.

Cite this