a) What is the smallest length n for which the Gilbert-Varshamov bound guarantees that there is a binary linear code of length n, dimension 4, and minimal distance 4? Let sucha code be called C.
(a) What is the smallest length n for which the Gilbert-Varshamov bound guarantees that there is a binary linear code of length n, dimension 4, and minimal distance 4? Let sucha code be called C.
(b) What is the smallest length m for which the Gilbert-Varshamov bound guarantees that there is a binary linear code of length m, dimension 4, and minimal distance 3? Let sucha code be called D.
(c) Let D' be obtained from D by adding a parity check to each codeword of D (so that thelength of D' is one more than the length of D, and the last bit of a codeword of D' is 1 ifthere is an odd number of 1s in the preceding positions, and 0 otherwise. What are the length, dimension, and minimal distance of D', and how does it compare to C?
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