Generalized MacWilliams identities and their applications to perfect binary codes
- Title
- Generalized MacWilliams identities and their applications to perfect binary codes
- Authors
- HYUN, JONG YOON; null
- Date Issued
- 2009-01
- Publisher
- SPRINGER
- 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 boolean OR w : z subset of 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.
- Keywords
- Generalized MacWilliams identities; Perfect binary codes
- URI
- https://oasis.postech.ac.kr/handle/2014.oak/29214
- DOI
- 10.1007/S10623-008-9
- ISSN
- 0925-1022
- Article Type
- Article
- Files in This Item:
- There are no files associated with this item.
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