Asymptotically Gaussian Weight Distribution and Performance of Multicomponent Turbo Block Codes and Product Codes

Asymptotically Gaussian Weight Distribution and Performance of Multicomponent Turbo Block Codes and Product Codes Dian-Wu Yue and En-hui Yang It was suggested by Battail that a good long linear code should have a weight distribution close to that of random coding, rather than a large minimum distance, and a turbo code should be also designed using a random-like criterion. In this paper, we first show that the weight distribution of a high-rate linear block code is approximately Gaussian if the code rate is close enough to one, and then proceed to construct a low-rate linear block code with approximately Gaussian weight distribution by using the turbo-coding technique. We give a sufficient condition under which the weight distribution of multicomponent turbo block (MCTB) codes (multicomponent product (MCP) codes, resp.) can approach asymptotically that of random codes, and further develop two classes of MCTB codes (MCP codes, resp.) satisfying this condition. Simulation results show that MCTB codes (MCP codes, resp.) having asymptotically Gaussian weight distribution can asymptotically approach Shannon\´s capacity limit. MCTB codes based on single parity-check (SPC) codes have a far poorer minimum distance than MCP codes based on SPC codes, but we show by simulation that when the bit-error rate is in the important range of and , these codes can still offer similar performance for the additive white Gaussian noise channel, as long as the code length of the SPC codes is not very short. These facts confirm, in a more precise way, Battail\´s inference about the “nonimportance” of the minimum distance for a long code.