Generalized MacWilliams identities and their applications to perfect binary codes

作者：Hyun Jong Yoon^*

来源：Designs, Codes and Cryptography, 2009, 50(2): 173-185.

DOI：10.1007/s10623-008-9222-6

• Generalized MacWilliams identities
• Perfect binary codes

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.

• 出版日期2009-2