New class of nonlinear systematic error detecting codes

A code C detects error e with probability 1-Q(e),ifQ(e) is a fraction of codewords y such that y, y+e∈C. We present a class of optimal nonlinear q-ary systematic (n, q^k)-codes (robust codes) minimizing over all (n, q^k)-codes the maximum of Q(e) for nonzero e. We also show that any linear (n, q^k)-code V with n ≤2k can be modified into a nonlinear (n, q^k)-code C_v with simple encoding and decoding procedures, such that the set E={e|Q(e)=1} of undetected errors for C_v is a (k-r)-dimensional subspace of V (|E|=q^k-r instead of q^k for V). For the remaining qⁿ-q^k-r nonzero errors, Q(e)≤q^-rfor q≥3 and Q(e)≤ 2^-r+1 for q=2.