Weight distribution of translates, covering radius, and perfect codes correcting errors of given weights

feat V be a binary linear (n, k) code defined by a check matrix H and let h(x) be the characteristic function for the set of columns of H . Connections between the Walsh transform of h(x) and the weight distributions of all translates of the code are obtained. Explicit formulas for the weight distributions of translates are given for small weights i(i < 8). The computation of the weight distribution of all translates (including the code itself) for i < 8 requires at most 7( n k)2”-k additions and subtractions, 6. 2”-k multiplications and 2”-k+’ memory cells. This method may be very effective if there is an analytic expression for h(x). A simple method for computing the covering radius of the code by the Walsh transform of h(x) is described. The implementation of this method requires for large n at most 2”-k (n k) logI (n k) arithmetical operations and 2”-kC’ memory cells. We define the concept L-perfect for codes, where L is a set of weights. After describing several linear and nonlinear L-perfect codes, necessary and sufficient conditions for a code to be L-perfect in terms of the Walsh transform of h(x) are established. An analog of the Lloyd theorem for such codes is proved.