Characteristic vector and weight distribution of a linear code

An Algorithm for Computing the Covering Radius of a Linear Code Based on Vilenkin-Chrestenson Transform

“…k is a θ(q, k) × q matrix. If we take v to be the characteristic vector χ = χ(C, G) of the linear code C with a generator matrix G, then the i-th coordinate of N (M k ) • χ T is equal to n − µ 0 where m is the i-th row of M k and µ 0 is the first coordinate of m [χ] (see [6]). The matrix M…”

“…k is used in the algorithm for calculating the weight distribution of a linear code with characteristic vector χ, presented in [6].…”

“…Remark 1 For given positive integers k and q (q is a prime power), the transformation r is defined for any integer valued vector v of length θ(q, k). As we will show below, the reduced distribution can be used both for computing the weight distribution of a linear code (see [6]), and for calculating the covering radius.…”

“…The size of the transformation vector in [5] is q k . In our study [6], in which we calculate the weight distribution of a linear code, we succeeded to reduce the length of the transformed vector to θ(q, k), using an appropriate new transformation. In this way, the number of required calculations is reduced by about q − 1 times.…”

A Search Method for Large Polarization Kernels

“…Equation ( 4) implies that to obtain all partial distances, one can compute the weight distribution of all kernel codes. A good review of weight distribution computation algorithms can be found in [16] together with a new method based on characteristic vector representation of linear codes.…”