DocumentCode :
914836
Title :
Weight enumerator for second-order Reed-Muller codes
Author :
Sloane, Neil J A ; Berlekamp, Elwyn R.
Volume :
16
Issue :
6
fYear :
1970
fDate :
11/1/1970 12:00:00 AM
Firstpage :
745
Lastpage :
751
Abstract :
In this paper, we establish the following result. Theorem: 
, the number of codewords of weight 
in the second-order binary Reed-Muller code of length 
is given by 
unless 
or 
, for some ![j, 0 \\leq j \\leq [m/2], A_0 = A_{2^m} = 1](/images/tex/7306.gif)
, and begin{equation} begin{split} A_{2^{m-1} pm 2^{m-1-j}} = 2^{j(j+1)} &{frac{(2^m - 1) (2^{m-1} - 1 )}{4-1} } \\ .&{frac{(2^{m-2} - 1)(2^{m-3} -1)}{4^2 - 1} } cdots \\ .&{frac{(2^{m-2j+2} -1)(2^{m-2j+1} -1)}{4^j -1} } , \\ & 1 leq j leq [m/2] \\ end{split} end{equation} begin{equation} A_{2^{m-1}} = 2 { 2^{m(m+1)/2} - sum_{j=0}^{[m/2]} A_{2^{m-1} - 2^{m-1-j}} }. end{equation}
, the number of codewords of weight in the second-order binary Reed-Muller code of length is given by unless or , for some , and begin{equation} begin{split} A_{2^{m-1} pm 2^{m-1-j}} = 2^{j(j+1)} &{frac{(2^m - 1) (2^{m-1} - 1 )}{4-1} } \\ .&{frac{(2^{m-2} - 1)(2^{m-3} -1)}{4^2 - 1} } cdots \\ .&{frac{(2^{m-2j+2} -1)(2^{m-2j+1} -1)}{4^j -1} } , \\ & 1 leq j leq [m/2] \\ end{split} end{equation} begin{equation} A_{2^{m-1}} = 2 { 2^{m(m+1)/2} - sum_{j=0}^{[m/2]} A_{2^{m-1} - 2^{m-1-j}} }. end{equation}Keywords :
Reed-Muller codes; Ash; Contracts; Convolution; Convolutional codes; Decoding; Equations; Error correction codes; Information theory; Polynomials; Telephony;
fLanguage :
English
Journal_Title :
Information Theory, IEEE Transactions on
Publisher :
ieee
ISSN :
0018-9448
Type :
jour
DOI :
10.1109/TIT.1970.1054553
Filename :
1054553
Link To Document :
