A weight formula for group codes

An explicit expression is given for the weights of any group code. The exponential weight of a vector, an element of a finite field, is introduced. It is a primitive root of unity raised to the weight of a vector. Using the general representation of group codes via polynomials, a formula is obtained for the exponential weight of a code word as a function of the parameters of the general representation. The coefficients of the formula are obtained using elementary multiplication on the cyclic group of the nth roots of unity. The formula is, in general, quite complicated and requires a computer for computation of its explicit form. However, for the new practical codes of a highly algebraic nature (Bose-Chaudhuri, Cyclic, Polynomial, Jump-Shift Register codes which are characterized by simple field parameters and polynomials), the formula emerges more pliable. It is hoped that use of the formula will yield more precise error-correcting properties of a given group code (minimum weight of ail non-zero vectors of a code) and the distribution of weights of a code.