Parity Check Matrix and Error Detection in Binary Codes, Assignments of Mathematics

Download Parity Check Matrix and Error Detection in Binary Codes and more Assignments Mathematics in PDF only on Docsity! M 375 – Homework 10 Page 1 M 375 – Homework 10 4.6.1 Parity check matrix H: = 101010101010101 110011001100110 111100001111000 111111110000000 H The columns of the parity check matrix are the binary representation of the number 1…15. (a) syndrome = 0001 first digit of codeword is wrong: correct codeword (000 000 000 000 000). (b) syndrome = 0000 no error: (111 111 111 111 111). 4.7.1 There are 8 possible binary codes of length 3. Those are { })111(),110(),101(),100(),011(),010(),001(),000( . But only 4 of them are linearly independent: a) (000): 0 b) (001): 1 c) (011): 1+x d) (111): 12 ++ xx All other codes can be produced by cyclic permutation of a) – d): (010) and (100): permute b) once and twice, respectively (101) and (110): permute c) once and twice, respectively. 4.7.2 Known from the lecture: 1)()( 7= xxhxg and 1)( 3 ++= xxxg . 1)1)(1( 242353457243 +++++++++++=+++++ xxxxxxxxxxxxxxxx . In ][F72 x this polynomial is reduced to: 1 7x .