On the weight spectrum of good linear binary codes

The weight spectrum of sequences of binary linear codes that achieve arbitrarily small word error probability on a class of noisy channels at a nonzero rate is studied. We refer to such sequences as good codes. The class of good codes includes turbo, low-density parity-check, and repeat-accumulate codes. We show that a sequence of codes is good when transmitted over a memoryless binary-symmetric channel (BSC) or an additive white Gaussian noise (AWGN) channel if and only if the slope of its spectrum is finite everywhere and its minimum Hamming distance goes to infinity with no requirement on its rate growth. The extension of these results to code ensembles in probabilistic terms follows in a direct manner. We also show that the sufficient condition holds for any binary-input memoryless channel.