A New Treatment of Bose-Chaudhuri Codes

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 gₐ (x) such that the coordinates of a are the values of gₐ (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 d₀, the usual Bose-Chaudhuri lower bound. This lower bound d₀ 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 9, thus improving by 4 the usual lower bound d₀ = 5 for that code.