Binary (18,11)2 codes do not exist-Nor do (64,53)2 codes

摘要

A binary, linear block code C with block length n and dimension n is commonly denoted by (n,k) or, if its minimum distance is d, by (n,k,d). The code's covering radius r(C) can be defined as the smallest number r such that any binary column vector of length (n-k) can be written as a sum of r or fewer columns of a parity-check matrix of C. An (n,k) code with covering radius r is denoted by (n,k)r. R.A. Brualdi et al., (1989) showed that l(m,r) is defined to be the smallest n such that an (n,n-m)r code exists. l(m,2) is known for m>or=6, while it is shown by Brualdi et al. that 17>or=l