Treffer: On binary codes with distances d and d+2.

We consider the problem of finding A 2 (n , { d 1 , d 2 }) defined as the maximal size of a binary (non-linear) code of length n with two distances d 1 and d 2 . Binary codes with distances d and d + 2 of size ∼ n 2 d 2 (d 2 + 1) can be obtained from 2-packings of an n-element set by blocks of cardinality d 2 + 1 . This value is far from the upper bound A 2 (n , { d 1 , d 2 }) ≤ 1 + n 2 proved recently by Barg et al. In this paper we prove that for every fixed d (d even) there exists an integer N(d) such that for every n ≥ N (d) it holds A 2 (n , { d , d + 2 }) = D (n , d 2 + 1 , 2) , where D (n , d 2 + 1 , 2) is the maximal size of a 2- (n , d 2 + 1 , 1) packing. In other words, an optimal code is a translation of some constant-weight code. We prove also estimates on N(d) for d = 4 and d = 6 . [ABSTRACT FROM AUTHOR]

Copyright of Designs, Codes & Cryptography is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)