Let h1(x)h2(x) be the parity check polynomial of a binary cyclic code. This article presents a formula for decomposing words in the code as sums of multiples of words in the codes whose parity check polynomials are h1(x) and h2(x). This decomposition provides information about the weight distribution of the code.
Hartmann, Carlos R.P.; Riek, J. R. Jr.; and Longobardi, Ralph J., "WEIGHT DISTRIBUTIONS OF SOME CLASSES OF BINARY CYCLIC CODES" (1974). Electrical Engineering and Computer Science Technical Reports. 20.