Parity Check Matrix Approach to Linear Block Codes
The parity check matrix can be assumed for an alternative description of a linear code. The relationship with the generator matrix is not univocal, except when such matrices are systematic. Upper triangular generator matrices and lower triangular parity check matrices are presented, showing the advantage of a clear recognition of information and control symbol positions. Error correction is outlined by means of properly processing single-error syndromes. A strict-sense time-invariant code in its parity check matrix is characterized by a unique parity check polynomial. The concept of dual code is discussed. Code puncturation and code shortening are interpreted as dual operations. Constant-length puncturation and constant-length shortening are described. Periodic parity check polynomials in lengthened cyclic codes are constructed. The parity check matrix of a modified lengthened cyclic code is derived. The difference between periodic and non-periodic parity check rows is stressed. H-extended cyclic codes and modified H-extended cyclic codes are introduced. Generalized repetition codes represent dual codes of generalized parity check codes. The whole family of Hamming codes is constructed by repeated lengthening operations. The whole family of simplex codes is constructed by repeated H-extensions. Direct product codes are described by means of their parity check matrix. An encoder circuit based on the parity check polynomial is depicted. Its state diagram is studied, and the trellis obtained from it is compared with the traditional generator trellis. Finally some considerations about the parity check matrix of nonbinary block codes are developed, with particular attention to their error correction capability.
KeywordsBlock Code LDPC Code Parity Check Code Word Cyclic Code
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