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Group codes outperform binary-coset codes on nonbinary symmetric memoryless channels

Como, Giacomo LU (2010) In IEEE Transactions on Information Theory 56(9). p.4321-4334
Abstract

Typical minimum distances and error exponents are analyzed on the 8-PSK Gaussian channel for two capacity- achieving code ensembles with different algebraic structure. It is proved that the typical group code over the cyclic group of order eight achieves both the Gilbert-Varshamov bound and the expurgated error exponent. On the other hand, the typical binary-coset codes (under any labeling) is shown to be bounded away both from the Gilbert-Varshamov bound (at any rate) and the expurgated exponent (at low rates). The reason for this phenomenon is shown to rely on the symmetry structure of the 8-PSK constellation, which is known to match the cyclic group of order eight, but not the direct product of three copies of the binary group. The... (More)

Typical minimum distances and error exponents are analyzed on the 8-PSK Gaussian channel for two capacity- achieving code ensembles with different algebraic structure. It is proved that the typical group code over the cyclic group of order eight achieves both the Gilbert-Varshamov bound and the expurgated error exponent. On the other hand, the typical binary-coset codes (under any labeling) is shown to be bounded away both from the Gilbert-Varshamov bound (at any rate) and the expurgated exponent (at low rates). The reason for this phenomenon is shown to rely on the symmetry structure of the 8-PSK constellation, which is known to match the cyclic group of order eight, but not the direct product of three copies of the binary group. The presented results indicate that designing group codes matching the symmetry of the channel guarantees better typical-code performance than designing codes whose algebraic structure does not match the channel. This contrasts the well-known fact that the average binary-coset code achieves both the capacity and the random-coding error exponent of any discrete memoryless channel.

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Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Coset codes, error exponent, expurgated exponent, Gilbert-Varshamov bound, group codes, linear codes, minimum distance, random codes
in
IEEE Transactions on Information Theory
volume
56
issue
9
article number
5550276
pages
14 pages
publisher
IEEE - Institute of Electrical and Electronics Engineers Inc.
external identifiers
  • scopus:77955740915
ISSN
0018-9448
DOI
10.1109/TIT.2010.2054330
project
Modeling and Control of Large Scale Transportation Networks
language
English
LU publication?
yes
id
b567b7b1-1242-4629-a540-cb731af84bf7
date added to LUP
2022-03-22 13:17:34
date last changed
2022-04-26 11:13:59
@article{b567b7b1-1242-4629-a540-cb731af84bf7,
  abstract     = {{<p>Typical minimum distances and error exponents are analyzed on the 8-PSK Gaussian channel for two capacity- achieving code ensembles with different algebraic structure. It is proved that the typical group code over the cyclic group of order eight achieves both the Gilbert-Varshamov bound and the expurgated error exponent. On the other hand, the typical binary-coset codes (under any labeling) is shown to be bounded away both from the Gilbert-Varshamov bound (at any rate) and the expurgated exponent (at low rates). The reason for this phenomenon is shown to rely on the symmetry structure of the 8-PSK constellation, which is known to match the cyclic group of order eight, but not the direct product of three copies of the binary group. The presented results indicate that designing group codes matching the symmetry of the channel guarantees better typical-code performance than designing codes whose algebraic structure does not match the channel. This contrasts the well-known fact that the average binary-coset code achieves both the capacity and the random-coding error exponent of any discrete memoryless channel.</p>}},
  author       = {{Como, Giacomo}},
  issn         = {{0018-9448}},
  keywords     = {{Coset codes; error exponent; expurgated exponent; Gilbert-Varshamov bound; group codes; linear codes; minimum distance; random codes}},
  language     = {{eng}},
  number       = {{9}},
  pages        = {{4321--4334}},
  publisher    = {{IEEE - Institute of Electrical and Electronics Engineers Inc.}},
  series       = {{IEEE Transactions on Information Theory}},
  title        = {{Group codes outperform binary-coset codes on nonbinary symmetric memoryless channels}},
  url          = {{http://dx.doi.org/10.1109/TIT.2010.2054330}},
  doi          = {{10.1109/TIT.2010.2054330}},
  volume       = {{56}},
  year         = {{2010}},
}