is called a subset if The all zeros vector is in S Sum of any two vectors in S

# Is called a subset if the all zeros vector is in s

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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 V i and V j are 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 coding 1. pack the largest amount of code vectors in the vector space to increase coding efficiency. 2. code vectors should be as far apart from each other as possible. Example: (6,3) linear block code Message Vector Code Vector 000 000 000 100 110 100 010 011 010 110 101 110 001 101 001 101 011 101 011 110 011 111 000 111 Only 8 out of 64, 6-tuples in the V 6 vector space are used as code vectors.
4.1.2.3 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 2 k member vectors of the subspace. The generating set of vectors is said to span the subspace. Code vector U is generated as a linear combination of k linearly independent n-tuples V 1 , V 2 , …, V k , i.e., Where m i =(0 or 1) are the message digits and i=1, 2,…, k
n general, a generator matrix is in the form of a (k n) matrix. If the message vector m is expressed as a row vector Then the code vector U is given by; U=mG Message vector Code vector Generator matrix
Example; Thus the code vector U corresponding to a message vector m is a linear combination of the rows of G . The encoder only needs to store the k rows of G instead of the total 2 k code vectors. Code vector Generator matrix
4.1.2.4 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 P is the parity array portion of the generator matrix with; p ij = (0 or 1), and I k is the k k identity matrix. Reduced complexity because it’s not necessary to store identity matrix.