Wide-Sense Time-Invariant Convolutional Codes in Their Parity Check Matrix
The circuit for syndrome calculation in a convolutional code is introduced. The most general method for obtaining the parity check matrix in a convolutional code, starting from its generator matrix, is presented. The minimal encoder circuit is discussed. Wide-sense time-invariant convolutional codes form a closed ensemble. Code puncturation is studied for convolutional codes looking at their parity check matrix. Tail-biting w.s. time-invariant convolutional codes are examined, considering both the generator and the parity check matrix. Unwrapping a quasi-cyclic code for obtaining convolutional codes, in general w.s. time-invariant, leads to a second conceptual bridge between these two classes of codes, in particular regarding their parity check matrix. This observation is important especially for low-density parity-check codes (LDPC codes). Array codes are now interpreted as LDPC block codes without short cycles of 1-symbols. It is possible to obtain an unwrapped form for such codes, in order to construct interesting LDPC convolutional codes. The equivalence between an array code with two component codes and a direct product code between two parity check codes is demonstrated. The concept of truncated circulants is transferred to a description based on the parity check matrix.
KeywordsLDPC Code Parity Check Convolutional Code Parity Check Matrix Component Code
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