NDLI: Generalized MacWilliams identities and their applications to perfect binary codes

Content Provider	Springer Nature Link
Author	Hyun, Jong Yoon
Copyright Year	2008
Abstract	We present generalized MacWilliams identities for binary codes. These identities naturally lead to the concepts of the local weight distribution of a binary code with respect to a word u and its MacWilliams u-transform. In the case that u is the all-one word, these ones correspond to the weight distribution of a binary code and its MacWilliams transform, respectively. We identify a word v with its support, and consider v as a subset of {1, 2,..., n}. For two words u,w of length n such that their intersection is the empty set, define the u-face centered at w to be the set $${\{z \cup w : z \subseteq u\}}$$ . A connection between our MacWilliams u-transform and the weight distribution of a binary code in the u-face centered at the zero word is presented. As their applications, we also investigate the properties of a perfect binary code. For a perfect binary code C, the main results are as follows: first, it is proved that our local weight distribution of C is uniquely determined by the number of codewords of C in the orthogonal u-face centered at the zero word. Next, we give a direct proof for the known result, concerning the weight distribution of a coset of C in the u-face centered at the zero word, by A. Y. Vasil’eva without using induction. Finally, it is proved that the weight distribution of C in the orthogonal u-face centered at w is uniquely determined by the codewords of C in the u-face centered at the zero word.
Ending Page	185
Page Count	13
Starting Page	173
File Format	PDF
ISSN	09251022
e-ISSN	15737586
Journal	Designs, Codes and Cryptography
Issue Number	2
Volume Number	50
Language	English
Publisher	Springer US
Publisher Date	2008-06-27
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Information and Communication, Circuits Data Structures, Cryptology and Information Theory Discrete Mathematics in Computer Science Perfect binary codes Data Encryption Linear codes, general Coding and Information Theory Generalized MacWilliams identities Combinatorics
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Discrete Mathematics and Combinatorics Theoretical Computer Science Computer Science Applications

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