DocumentCode :
917496
Title :
Gleason´s theorem on self-dual codes
Author :
Berlekamp, Elwyn R. ; MacWilliams, F. Jessie ; Sloane, Neil J A
Volume :
18
Issue :
3
fYear :
1972
fDate :
5/1/1972 12:00:00 AM
Firstpage :
409
Lastpage :
414
Abstract :
The weight enumerator of a code is the polynomial begin{equation} W(x,y)= sum_{r=0}^n A_r x^{n-r} y^r, end{equation} where 
denotes the block length and 
, denotes the number of codewords of weight 
. Let 
be a self-dual code over 
in which every weight is divisible by 
. Then Gleason\´s theorem states that 1) if 
= 2 and = 2, the weight enumerator of is a sum of products of the polynomials 
and 
if = 2 and = 4, the weight enumerator is a sum of products of 
and 
; and 3) if = 3 and = 3, the weight enumerator is a sum of products of 
and 
. In this paper we give several proofs of Gleason\´s theorem.
denotes the block length and , denotes the number of codewords of weight . Let be a self-dual code over in which every weight is divisible by . Then Gleason\´s theorem states that 1) if = 2 and = 2, the weight enumerator of is a sum of products of the polynomials and if = 2 and = 4, the weight enumerator is a sum of products of and ; and 3) if = 3 and = 3, the weight enumerator is a sum of products of and . In this paper we give several proofs of Gleason\´s theorem.Keywords :
Dual codes; Decoding; Error correction codes; Information theory; Notice of Violation; Polynomials;
fLanguage :
English
Journal_Title :
Information Theory, IEEE Transactions on
Publisher :
ieee
ISSN :
0018-9448
Type :
jour
DOI :
10.1109/TIT.1972.1054817
Filename :
1054817
Link To Document :
