Consider the following linear code. C= {0000, 0010, 0101, 0111, 1001, 1011, 1100, 1110} (You do...

  

Transcribed Image Text:

Consider the following linear code. C= {0000, 0010, 0101, 0111, 1001, 1011, 1100, 1110} (You do not need to prove that this code is linear.) (a) Find the barameters n, M and d of this code. (b) By finding a binary string of length 4 (that is not a codeword of C) which has at least two nearest neighbours in C, prove that C is not a perfect code. (c) Find a basis B for the code C. Demonstrate that B is a basis by showing that every codeword of C is either an element of B, or can be obtained by adding some or all of the elements of B, or by adding an element of B to itself. (Hint: Any basis of C will contain three elements.) (d) Use your basis from part (c) to form a generator matrix for C, and use your generator matrix to generate at least four of the codewords of C. Consider the following linear code. C= {0000, 0010, 0101, 0111, 1001, 1011, 1100, 1110} (You do not need to prove that this code is linear.) (a) Find the barameters n, M and d of this code. (b) By finding a binary string of length 4 (that is not a codeword of C) which has at least two nearest neighbours in C, prove that C is not a perfect code. (c) Find a basis B for the code C. Demonstrate that B is a basis by showing that every codeword of C is either an element of B, or can be obtained by adding some or all of the elements of B, or by adding an element of B to itself. (Hint: Any basis of C will contain three elements.) (d) Use your basis from part (c) to form a generator matrix for C, and use your generator matrix to generate at least four of the codewords of C.