---
res:
bibo_abstract:
- "We describe a new method to compare the bit-MAP and block-MAP decoding thresholds
of Reed-Muller (RM) codes for transmission over a binary memoryless symmetric
channel. The question whether RM codes are capacity-achieving is a long-standing
open problem in coding theory and it has recently been answered in the affirmative
for transmission over\r\nerasure channels. Remarkably, the proof does not rely
on specific properties of RM codes, apart from their symmetry. Indeed, the main
technical result consists in showing that any sequence of linear codes, with doubly-transitive
permutation groups, achieves capacity on the memoryless erasure channel under
bit-MAP decoding. A natural question is what happens under block-MAP decoding.
If the minimum distance of the code family is close to linear (e.g., of order
N/ log(N)), then one can combine an upper bound on the bit-MAP error probability
with a lower bound on the minimum distance to show that the code family is also
capacity-achieving under block-MAP decoding. This strategy is successful for BCH
codes. Unfortunately, the minimum distance of RM codes scales only as √N, which
does not suffice to obtain the desired result. Then, one can exploit further symmetries
of RM codes to show that the bit-MAP threshold is sharp enough so that the block
erasure probability also tends to 0. However, this technique relies heavily on
the fact that the transmission is over an erasure channel.\r\nWe present an alternative
approach to strengthen results regarding the bit-MAP threshold to block-MAP thresholds.
This approach is based on a careful analysis of the weight distribution of RM
codes. In particular, the flavor of the main result is the following: assume that
the bit-MAP error probability decays as N−δ, for some δ > 0. Then, the block-MAP\r\nerror
probability also converges to 0. This technique applies to the transmission over
any binary memoryless symmetric channel. Thus, it can be thought of as a first
step in extending the proof that RM codes are capacity-achieving to the general
case.@eng"
bibo_authorlist:
- foaf_Person:
foaf_givenName: Marco
foaf_name: Mondelli, Marco
foaf_surname: Mondelli
foaf_workInfoHomepage: http://www.librecat.org/personId=27EB676C-8706-11E9-9510-7717E6697425
orcid: 0000-0002-3242-7020
- foaf_Person:
foaf_givenName: Shrinivas
foaf_name: Kudekar, Shrinivas
foaf_surname: Kudekar
- foaf_Person:
foaf_givenName: Santosh
foaf_name: Kumar, Santosh
foaf_surname: Kumar
- foaf_Person:
foaf_givenName: Henry D.
foaf_name: Pfister, Henry D.
foaf_surname: Pfister
- foaf_Person:
foaf_givenName: Eren
foaf_name: Şaşoğlu, Eren
foaf_surname: Şaşoğlu
- foaf_Person:
foaf_givenName: Rüdiger
foaf_name: Urbanke, Rüdiger
foaf_surname: Urbanke
bibo_doi: 10.3929/ETHZ-A-010646484
dct_date: 2016^xs_gYear
dct_language: eng
dct_publisher: ETH Zürich@
dct_title: 'Reed-Muller codes: Thresholds and weight distribution@'
...