Optimal binary constant weight codes and affine linear groups over finite fields

  • Xiang-Dong HouEmail author


The affine linear group of degree one, \(\text {AGL}(1,\mathbb {F}_q)\), over the finite field \(\mathbb {F}_q\), acts sharply two-transitively on \(\mathbb {F}_q\). Given \(S<\text {AGL}(1,\mathbb {F}_q)\) and an integer k, \(1\le k\le q\), does there exist a k-element subset \(B\subset \mathbb {F}_q\) whose set-wise stabilizer is S? Our main result is the derivation of two formulas which provide an answer to this question. This result allows us to determine all possible parameters of binary constant weight codes that are constructed from the action of \(\text {AGL}(1,\mathbb {F}_q)\) on \(\mathbb {F}_q\) to meet the Johnson bound. Consequently, for many parameters, we are able to determine the values of the function \(A_2(n,d,w)\), which is the maximum number of codewords in a binary constant weight code of length n, weight w and minimum distance \(\ge d\).


Affine linear group BIBD Constant weight code Johnson bound 

Mathematics Subject Classification

05B05 05E18 94B25 



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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of South FloridaTampaUSA

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