LDPC Convolutional Codes

Abstract

The best results on LDPC convolutional codes are summarized. Such results have been obtained, up to now, after proper transformations of good LDPC block codes. Unwrapping assures better performance with respect to that of the original parent block code. Irregular codes appear to be preferable. Modified H-extension offers good possibilities starting from BPG block codes. Nevertheless the parity check constraint length is extremely high. The direct design of LDPC convolutional codes with very short constraint length is outlined, starting from the case of code rate 1/2. A parity check matrix not in minimal form is typically employed, especially when the code is regular. Irregular codes allow to obtain good performance with shorter constraint lengths. The case of high-rate convolutional code is treated deriving their properties from those of 1/2-rate codes. Low-rate convolutional codes require a more involved treatment, since some vertical separations between 1-symbols in their syndrome former sub-matrix can be reused. The case of period length and number of information symbols per period not co-prime is analyzed. Some interesting 2/4-rate codes are presented. Regular and irregular codes are compared, fixing the code rate and minimum distance, in search for the minimum parity check constraint length. Hints for a direct design of LDPC convolutional codes with short constraint length are given. The concept of doubly convolutional codes is revisited.