Weight distribution of low-density periodic random errors and their correcting codes with error decoding probability

We present weight distribution of low-density periodic random errors in the space of all $q$-ary $n$-tuples along with the average Hamming weight of the error set. We also provide necessary and sufficient conditions for the existence of linear codes correcting such error pattern. Examples of such codes are given. Finally, probability of decoding error of such codes over a binary symmetric channel is derived.