linear codes
1. discuss homework.
2. go over material in chapter 8 from text :
a) "linear code"
b) parameters : (n, k, delta, info rate, |C|)
c) "check matrix"
Discuss in terms of linear algebra :
- N x K "coding matrix" = basis vectors as columns
- (N-K) x N "check matrix" = orthogonal "kernel" as rows; defines same space
- codewords are vectors spanned by basis
- bits of info = k = independent "components" (each in or out) of code words
In this "binary" arithmetic :
- twice anything is zero
- ... which gives simple way to connect "coding" with "check" matrix
- ... when both are in "standard form"
- Also gives a simple trick for finding where a single error happened
- ... at least for for 1-error correcting codes
Do several examples out by hand, constructing
- the coding matrix
- the check matrix
- giving n, k, delta, info rate
- code words
- encoding a message
- correcting an error (for 1-error correcting code)
