LCD Codes and Iterative Decoding by Projections, a First Step Towards an Intuitive Description of Iterative Decoding

Jalal Etesami, Fangning Hu, Werner Henkel

Abstract: From our earlier works, we know that in the case of analog codes, a Turbo-like iterative decoding can be nicely illustrated as iterative projections onto super codes that correspond to parts of the parity check matrix. So-called LCD (linear code with complementary dual) codes are recognized as a counterpart in finite fields for the orthogonal case, where two iterative projections lead to the final solution. A method for decomposing an arbitrary LCD code C into two super LCD codes C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> and C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> such that decoding by iteratively projecting the received vector onto C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> and C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> results in the same decoding solution as directly projecting the vector onto the original code space C. This is not necessarily a maximum-likelihood solution opposite to the analog case. A bound on the probability of finding the nearest codeword is provided.