Strict-Sense Time-Invariant Convolutional Codes in Their Parity Check Matrix
The syndrome former sub-matrix is introduced. The generator matrix of a convolutional code can be constructed by means of a procedure called null ex-OR sum of clusters of syndromes. Inversely, it is possible to operate by means of the column construction of the parity check matrix. The importance of a parity check matrix in minimal form is stressed. A strict-sense time-invariant convolutional code in its parity check matrix is characterized by a unique interleaved parity check polynomial. It is possible to distinguish between low-rate and high-rate convolutional codes, taking into account their parity check matrix. The dual of a low-rate convolutional code is a high-rate convolutional code, but the outermost parts of the frame show different structures in the two matrices. A systematic encoder circuit based on the unique interleaved parity check polynomial is presented. Traditional encoder circuits for convolutional codes described by means of the parity check matrix are discussed, considering a unique shift register in observer arrangement. Not well designed convolutional codes and the presence of periodic rows in their parity check matrix are studied. The tail-biting arrangement of a convolutional code is revisited looking at its parity check matrix. When the code is not well-designed, the tail-biting convolutional code has a generator matrix not of full rank, and its periodic row in the parity check matrix vanishes. A second conceptual bridge between cyclic block codes and convolutional codes is presented, now based on the parity check matrix. Modified lengthening in the generator matrix and H-extension in the parity check matrix support this correspondence. A family tree for many classes of error correcting codes is depicted. Not well-designed convolutional codes can be considered the dual with respect to convolutional codes whose parity check matrix is not in its minimal form.
KeywordsCode Rate LDPC Code Parity Check Cyclic Code Convolutional Code
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