Course Intended Learning OutcomesExplain the basic concepts of error detection/correction coding and perform error analysis4.1Block Codes4.2Cyclic Codes
4.1.1Parity-check code1) Single-Parity-check-code•Redundant bit = 1 bit even or odd parity•Code rate k/(k+1)•Can only detect only odd number of bit errors, cannot correct errors•Probability of undetected error Pud(even number of bits are inverted)p is the bit error probability
MessageParityCodeword0000000010011100010110101100011000111001101001010110001111111111Example: Even-Parity Code•A (4, 3) even parity, error-detection code•Probability of channel symbol error is; •Can detect single or triple error patterns.•Probability of undetected error is equal to the probability that two or four errors occur anywhere in a codeword.
2) Rectangular Code (Product Code)•Can be considered as a parallel data transmission.•Data bloc; M rows and N columns•Append•a horizontal parity check to each row and•a vertical parity check to each column•Coded block; (M+1) rows and (N+1) columns•Can correct single bit error•Serial transmission;111001000011110000111100110110Verticalparity checkHorizontalparity check111001000011110000111100110110
4.1.2Linear Block Codes•Belong to a class of parity check codes characterized by (n, k) notation•Encoder transforms a block of k message digits (a message vector) into a longer block of n codeword digits (a code vector) constructed from a given alphabet of elements•2kk-tuples (sequence of k digits) messages are linearly, uniquely mapped to 2kn-tuples codewords•This mapping is accomplished via a look-up table.
126.96.36.199 Vector Spaces•Set of all binary n-tuples Vnis called a vector space over the binary field of two elements (0 and 1).•The field has two operations, addition (XOR) and multiplication (AND).AdditionMultiplication188.8.131.52 Vector SubspacesSubset S of the vector space Vn