Abstract
Certain classes of binary constant weight codes can be represented geometrically using linear structures in Euclidean space. The geometric treatment is concerned mostly with codes with minimum distance 2(w − 1), that is, where any two codewords coincide in at most one entry; an algebraic generalization of parts of the theory also applies to some codes without this property. The presented theorems lead to several improvements of the tables of lower bounds on A(n, d, w) maintained by E. M. Rains and N. J. A. Sloane, and the ones recently published by D. H. Smith, L. A. Hughes and S. Perkins. Some of these new codes can be proven optimal.
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References
Brouwer AE, Shearer JB, Sloane NJA, Smith WD (1990) A new table of constant weight codes. IEEE Trans Inf Theory 36:1334–1380
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Rains EM, Sloane NJA. Table of constant weight binary codes. http://www.research.att.com/njas/codes/Andw/
Smith DH, Hughes LA, Perkins S (2006) A new table of constant weight codes of length greater than 28. Electron J Comb 13:1–18
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Gashkov, I., Ekberg, J.A.O. & Taub, D. A geometric approach to finding new lower bounds of A(n, d, w). Des Codes Crypt 43, 85–91 (2007). https://doi.org/10.1007/s10623-007-9064-7
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DOI: https://doi.org/10.1007/s10623-007-9064-7