A New Treatment of Bose-Chaudhuri Codes
Harold F. Mattson, G. Solomon · Journal of the Society for Industrial and Applied Mathematics · 1961
Letting A be any $(k,n)$ Bose-Chaudhuri code, we first attach to each a in A (via difference equations over $GF(2)$) a polynomial $g_a (x)$ such that the coordinates of a are the values of $g_a (x)$ on the nth roots of unity. The degree of these polynomials is such that the minimum nonzero weight d of vectors in A is immediately seen to be at least $d_0 $, the usual Bose-Chaudhuri lower bound. This lower bound $d_0 $ is improved over a class of $(h + 1,p)$ codes, where $p = 2h + 1$ has certain prime values, in a number of general theorems. In particular, the (12, 23) Golay code is proved very simply to have $d = 7$; and a (24, 47) code is shown to have $d\leqq 9$, thus improving by 4 the usual lower bound $d_0 = 5$ for that code.