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A geometric approach to finding new lower bounds of $A(n,d,w)$. (English)
Des. Codes Cryptography 43, No. 2-3, 85-91 (2007).
The largest possible size of a binary code of length $n$, constant weight $w$, and minimum Hamming distance $d$ is denoted by $A(n,d,w)$. Lower bounds on $A(n,d,w)$ can be obtained by constructing corresponding codes. One (trivial) family of good constant weight codes can be obtained by taking the codewords as the columns of a $v(v-1)/2 \times v$ 0-1 matrix where each row contains exactly two 1s and no two rows are identical; this code attains $A(v(v-1)/2,2v-4,v-1)=v$. The authors find lower bounds on and exact values of $A(n,2w-2,w)$ by a construction starting from the aforementioned (and similar) codes. The new bounds obtained are for codes of length greater than 28, which have only received limited interest in the literature, so it is difficult to say how the constructions presented compare with other known methods.
Reviewer: Patric R. J. Östergård (Helsinki)