Some results on the weight distribution of linear codes

By studying the algebraic structure of the parity check matrix of a linear code we show that the weight distribution is a function only of the quantities
N_{\upsilon}^{u}
, the number of
\upsilon
columns of the parity check matrix with rank
u
. We apply this to obtain a new formula for the weight distribution of the distance
3 s
-ary Hamming code and for the distance
4 s
-ary conic code. We give the definition of a conic code and some of its properties.