Parity Check Matrix
Contents
Encoding
- Write the codeword as x1 x2 x3 x4 … where needed, substitute in the message
- Multiply H by the codeword, equate each row to 0
- Solve to find the check-bits
- Merge
Decoding
- Multiply H by the message (as a vector)
- If the resulting vector is
- Zero vector - no errors, continue
- Non-zero - Fix the bit associated with the index of the resulting vector in the parity matrix
Strip check-bits to extract message
**If the resulting vector is a multiple of one of the columns in the parity matrix, substract the related bit
n
times, such that the resulting vector divides the parity columnn
times.
Codewords in a ternary generator matrix: 3^n rows
Codewords in the basis of a binary linear code with parity check matrix H. (minimum number of elements needed to create all other) (h without the I matrix?)
Greatest number of information bits for a radix 3 2 error linear code with m=5 check bits.
Given a parity check matrix H, the minimum distance d© of the code is the minimal weight.
Minimal (Hamming) distance = minimum number of changes == smallest number of linearly dependent columns
Read lecture three