Generator polynomial Consider nonzero codeword of a cyclic code having the smallest degree of its polynomial, i.e. having the longest all-zero sequence at first (leftmost) bits: . For the considered cyclic code (7,4), defined by parity-check matrix (7.1.1), : . Each cyclic shift of this word is als...
Hamming code 7 4 is the first code in the class of Hamming codes. Error syndromes, corresponding to single-error patterns, are written as binary words: that encode a position of nonzero bit in z , as presented in Table i 1 1 001 2 01 010 3 001 011 4 0001000 100 5 100 101 6 10 110...
I NTRODUCTION Term of information is in our civilization so broad that it is impossible to expect a definition of the universal measure of information. In many domains, especially in (tale)communication, source of information, normally discrete data, and transmission channel are described by their ...
Kraft inequality If integers satisfy: then, a prefix condition code of alphabet length D exists, with these integers as codeword lengths. A convenient graphical representation of a code is a tree built of branches connecting nodes, with sequences assigned to nodes, codewords (sequences satisfying...
Markov source A discrete memoryless source generates successive messages (symbols) randomly, independently of the source history. Such source probabilistic description is dependent of a time origin and is called non-stationary. A discrete source is...
Measurement strategy From all possible combinations of source data involved in the measurement/test, select such combination that gives the maximum mutual information, minimum information loss at the same time. For the optimum combination, all results should be equally probable (see P1), Repeat thi...
Measurement test program development problem statement Measuring device gives possible readings and it is assumed that measurement/test is performed with no misinformation, , i.e. . The measurement/test has to be repeated k times, to fully compensate a source entropy by the acquired mutual informat...
Mutual information of Discrete Memoryless Channel (DMC) For the given, set of input messages : , set of output messages : , Discrete Memoryless Channel (DMC) is defined by its probabilistic model (Fig.3.1), MN transition (conditional) probabiliti...
Optimum encoding (Huffman Code) For any given source with elements (messages), the optimum binary code exists, in which the two least likely code words and have the same length and differ in only the last bit, ending in a 0 and ending in a 1. Shorter codewords encode more likely source elements, i...
Parity check matrix An ( n , k ) linear code can be uniquely defined by a system of m = n k linear equations that express parity check bits by the information bits. It is normally assumed that weight of each codeword is even. Such code is called the even-parity block code. The parity-check matrix...
