Exercise 522

Exercise 522 In Exercise 431 , you were asked to show that there are t=(qr1)/(q1) (r1)-dimensional subcodes of an r-dimensional code over Fq. Do the following: (a) Prove that a binary code of dimension k+1 containing the all-one vector 1 has 2k1 subcodes of dimension k also containing 1. (b) Prove that a binary code of dimension k+1 containing an [n,m] subcode C has 2k+1m1 subcodes of dimension k also containing C. and the lemma follows. Exercise 431 Prove that there are t=(qr1)/(q1)(r1)-dimensional subcodes of an r-dimensional code over Fq. Theorem 7.10.10 (Generalized Griesmer Bound) Let C be an [n,k] code over Fq. Then for 1rk, ndr(C)+i=1krqi(qr1)q1dr(C). Proof: If r=k, the inequality reduces to the obvious assertion ndk(C); see Theorem 7.10.1. Now assume that r