Reed-Muller Codes Achieve Capacity on Erasure Channels

We introduce a new approach to proving that a sequence of deterministic linear codes achieves capacity on an erasure channel under maximum a posteriori decoding. Rather than relying on the precise structure of the codes, our method exploits code symmetry. In particular, the technique applies to any...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
Veröffentlicht in:IEEE transactions on information theory Jg. 63; H. 7; S. 4298 - 4316
Hauptverfasser: Kudekar, Shrinivas, Kumar, Santhosh, Mondelli, Marco, Pfister, Henry D., Sasoglu, Eren, Urbanke, Rüdiger L.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York IEEE 01.07.2017
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Schlagworte:
Affine-invariant codes
BCH codes
Binary codes
Boolean algebra
Boolean functions
capacity-achieving codes
Codes
Computers
Convergence
Decoding
Electronic mail
erasure channels
EXIT functions
Information transfer
Linear codes
MAP decoding
monotone Boolean functions
Parity check codes
Permutations
quadratic-residue codes
Reed-Muller codes
Symmetry
Transfer functions
ISSN:0018-9448, 1557-9654
Online-Zugang:Volltext
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
Beschreibung
Zusammenfassung:We introduce a new approach to proving that a sequence of deterministic linear codes achieves capacity on an erasure channel under maximum a posteriori decoding. Rather than relying on the precise structure of the codes, our method exploits code symmetry. In particular, the technique applies to any sequence of linear codes where the blocklengths are strictly increasing, the code rates converge, and the permutation group of each code is doubly transitive. In other words, we show that symmetry alone implies near-optimal performance. An important consequence of this result is that a sequence of Reed-Muller codes with increasing block length and converging rate achieves capacity. This possibility has been suggested previously in the literature but it has only been proven for cases where the limiting code rate is 0 or 1. Moreover, these results extend naturally to all affine-invariant codes and, thus, to extended primitive narrow-sense BCH codes. This also resolves, in the affirmative, the existence question for capacity-achieving sequences of binary cyclic codes. The primary tools used in the proof are the sharp threshold property for symmetric monotone Boolean functions and the area theorem for extrinsic information transfer functions.
Bibliographie:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2017.2673829