Weight distribution of translates, covering radius, and perfect codes correcting errors of given weights
Mark G. Karpovsky · IEEE Transactions on Information Theory · 1981
LetVbed binary linear(n,k)code defined by a check matrixHand leth(x)be the characteristic function for the set of columns ofH. Connections between the Walsh transform ofh(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 weightsi(i<8). The computation of the weight distribution of all translates (including the code itself) fori<8requires at most7(n-k)2^{n-k}additions and subtractions,6 \cdot 2^{n-k}multiplications and2^{n-k+l}memory cells. This method may be very effective if there is an analytic expression forh(x). A simple method for computing the covering radius of the code by the Walsh transform ofh(x)is described. The implementation of this method requires for largenat most2^{n-k}(n-k) \log_{2}(n-k)arithmetical operations and2^{n-k+1}memory cells. We define the conceptL-perfect for codes, whereLis a set of weights. After describing several linear and nonlinearL-perfect codes, necessary and sufficient conditions for a code to beL-perfect in terms of the Walsh transform ofh(x)are established. An analog of the Lloyd theorem for such codes is proved.