مرکز منطقه ای اطلاع رساني علوم و فناوري - Weight distribution of translates, covering radius, and perfect codes correcting errors of given weights

Abstract :

Let $V$ bed 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 $7(n-k)2^{n-k}$ additions and subtractions, $6 \\cdot 2^{n-k}$ multiplications and $2^{n-k+l}$ 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 $2^{n-k}(n-k) \\log _{2}(n-k)$ arithmetical operations and $2^{n-k+1}$ 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.