is called a subset if;•The all-zeros vector is in S.•Sum of any two vectors in S is also in S (closure property)Suppose Viand Vjare two codewords (or code vectors) in an (n, k) binary block code. The code is said to be linear if () is also a code vector. (property of linear dependence)
4.0000000100100011010001010110011110001001101010111100110111101111An example of subspace of V4contains;0000010110101111Add any two of this subspace vectors.What do you get?SKLAR Fig. 6.10
Goals for coding1.pack the largestamount of code vectors in the vector space to increase coding efficiency.2.code vectors should be as farapart from each other as possible.Example: (6,3) linear block codeMessage VectorCode Vector000000 000100110 100010011 010110101 110001101 001101011 101011110 011111000 111Only 8 out of 64, 6-tuples in the V6vector space are used as code vectors.
22.214.171.124 Generator Matrix•When k is large, a look-up table is not feasible, instead a generator matrix is used.•For a k-dimensional subspace of the n-dimensional binary vector space (k<n),•A set of n-tuples can generate all the 2kmember vectors of the subspace.•The generating set of vectors is said to span the subspace.•Code vector Uis generated as a linear combination of k linearly independent n-tuples V1, V2, …, Vk,i.e.,Where mi=(0 or 1) are the message digits and i=1, 2,…, k
n general, a generator matrix is in the form of a (kn) matrix.If the message vector mis expressed as a row vectorThen the code vector Uis given by; U=mGMessage vectorCode vectorGenerator matrix
Example;•Thus the code vector Ucorresponding to a message vector mis a linear combination of the rows of G.•The encoder only needs to store the k rows of Ginstead of the total 2kcode vectors.Code vectorGenerator matrix
126.96.36.199 Systematic Linear Block Codes•A systematic (n, k) linear block code is a mapping.•Mapping from k-dimensional message vector to a n-dimensional code vector.•It is done in such a way that part of the sequence generated coincides with the k message digits.•The remaining (n-k) digits are parity bits.Where Pis the parity array portion of the generator matrix with;pij= (0 or 1), and Ikis the kk identity matrix.Reduced complexity because it’s not necessary to store identity matrix.