Strict-Sense Time-Invariant Convolutional Codes in Their Parity Check Matrix

Abstract

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.