The sphere packing bound by Shannon [18] provides a lower bound to the frameerror rate (FER) achievable by an(n,k,d)code but is not directly applicable tobinary codes. Gallager [4] presented his coding theorem for the average FER forthe ensemble of all random binary(n,k,d)codes. There are 2npossible binarycombinations for each codeword which in terms of then-dimensional signal spacehyp…