28-11-2012, 05:52 PM
Linear Block Codes
5 Linear Block Codes.ppt (Size: 767.5 KB / Downloads: 162)
Basic Definitions
m is a k-bit information sequence
c is an n-bit codeword.
is a bit-by-bit mod-2 addition without carry
Linear code: The sum of any two codewords is a codeword.
Observation: The all-zero sequence is a codeword in every
Basic Definitions
Def: The weight of a codeword ci , denoted by w(ci), is the
number of of nonzero elements in the codeword.
Def: The minimum weight of a code, wmin, is the smallest
weight of the nonzero codewords in the code.
Theorem: In any linear code, dmin = wmin
Linear Block Codes
the number of codeworde is 2k since there are 2k distinct messages.
The set of vectors {gi} are linearly independent since we must have a set of unique codewords.
linearly independent vectors mean that no vector gi can be expressed as a linear combination of the other vectors.
These vectors are called baises vectors of the vector space C.
The dimension of this vector space is the number of the basis vector which are k.
Generator Matrix
All 2k codewords can be generated from a set of k linearly independent codewords.
The simplest choice of this set is the k codewords corresponding to the information sequences that have a single nonzero element.