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 quantitiesN_{\upsilon}^{u}, the number of\upsiloncolumns of the parity check matrix with ranku. We apply this to obtain a new formula for the weight distribution of the distance3 s-ary Hamming code and for the distance4 s-ary conic code. We give the definition of a conic code and some of its properties.