Exercise 29

Basic concepts of linear codes As in (ii), (iii) is completed if we show that CT has dimension k-1 IS CT with 5 of size d - 1. C has dimension k by part (ti). Clearly has minimum distance 1 and is obtained by pancturing on the nonzero coordinate of a weight I codeword in C. By Theorem 1.5.1(ii) c' has dimension - 1. Exercise 28 LetC be the binary repetition code of length as described in Example 1.2.2. Describe (C) and (Cr) for any T. Exercise 29 LetC be the code of length 6 in Example 14.4. Give generator matrices for (C) and (Cy) when 7 - 11.2) and 7 = {1,3). 6.4 Direct sums Fori 11.2) let C be an (n.di.di] code, both over the same finite tield F. Then their direct son is the Instna, ktk mind..dall code ciecz.2) CICEC). If C, has generator matrix G, and parity check matrix H. then TH GG, and HH (1.4) 0 H are a generator matrix and purity check matrix for Ciec Exercise 30 Let C have generator matrix and parity check matrix Hi fori 1.2). Prove that the generator and purity check matrices for CC, are as given in (14) . Exercise 31 Let be the binary code with generator matrix 1 100110 1 0 1 0 1 0 1 G-1001 110 1 0 1 0 1 1 0 1 0 0 1 0 1 1 Give another generator matrix for that shows that is a direct sum of two binary codes Example 1.5.8 The [6.3.2] binary code of Example 1.4.4 is the direct sum DDD of the 12.1.2] code D=100.11). Since the minimum distance of the direct sum of two codes does not exceed the minimum distance of cither of the codes, the direct sum of two codes is generally of little use in applications and is primarily of theoretical interest 1.5.5 The (u + v) construction Two codes of the same length can be combined to form a third code of twice the length in a way similar to the direct sum construction. Let C, be an in. dil code for 1 (1.2). 0:01