5. The generator matrix of an (n, k) linear block code (LBC) is given by: 1 1 1

Question

5. The generator matrix of an (n, k) linear block code (LBC) is given by: 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 1 0 1 0 1 0 1 Determine the codeword length n, information word length (dimensionality) k, code rate, and overhead of this LBC. Determine the minimum distance of the code. What is the error correction capability of the code? What is the error detection capability of the code? Transform the generator matrix into systematic form and provide the parity-check matrix of this code. Graduate students only: Provide the bipartite graph of this parity-check matrix and determine the corresponding girth.

Explanation / Answer

5) From the generator matrix we can tell code word length is 8(no of coloumns of G), and information word length is 4 (no of rows). and Over head of this LBC is 8-4 =4.

The total possible code words are 2^4 information word length.

From the 16 code words we need to caliclate minmum distance between codewords genrated from G.

All possible code words has minimum distance is 4.

Rearranging G is formed into as follows

1 0 0 0 1 0 1 1

0 1 0 0 0 1 1 1

0 0 1 0 0 0 1 1

0 0 0 1 1 1 0 1

From this parity matrix p as follows

1 0 1 1

0 1 1 1

0 0 1 1

1 1 0 1

parity check matrix is [pT: I n-k]

1 0 0 1 1 0 0 0

0 1 0 1 0 1 0 0

1 1 1 0 0 0 1 0

1 1 1 1 0 0 0 1

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