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Designs, Codes and Cryptography

pp 1–14

First Online: 20 May 2017Received: 17 June 2016Revised: 17 March 2017Accepted: 06 May 2017DOI: 10.1007-s10623-017-0366-0

Cite this article as: Polak, S.C. Des. Codes Cryptogr. 2017. doi:10.1007-s10623-017-0366-0


For \q,n,d \in \mathbb {N}\, let \A qn,d\ be the maximum size of a code \C \subseteq q^n\ with minimum distance at least d. We give a divisibility argument resulting in the new upper bounds \A 58,6 \le 65\, \A 411,8\le 60\ and \A 316,11 \le 29\. These in turn imply the new upper bounds \A 59,6 \le 325\, \A 510,6 \le 1625\, \A 511,6 \le 8125\ and \A 412,8 \le 240\. Furthermore, we prove that for \\mu ,q \in \mathbb {N}\, there is a 1–1-correspondence between symmetric \\mu ,q\-nets which are certain designs and codes \C \subseteq q^{\mu q}\ of size \\mu q^2\ with minimum distance at least \\mu q - \mu \. We derive the new upper bounds \A 49,6 \le 120\ and \A 410,6 \le 480\ from these ‘symmetric net’ codes.

KeywordsCode Nonbinary code Upper bounds Kirkman system Divisibility Symmetric net Communicated by V. A. Zinoviev.

Mathematics Subject Classification94B65 05B30 

Autor: Sven C. Polak


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