By Paul P. Martin

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E. a subset of Fqn for some n) generated by G, we have a natural identification between C and Fqk (k = dim C, not the same as n, the length of the code). Each x ∈ C is uniquely expressible as k x= ai vi i=1 62 CHAPTER 3. CODING THEORY (the vi s are the rows of G in the natural order). , ak ) ∈ Fqk is a one-to-one correspondence. , ak ) vectors as the message words of the code, and the n-tuples x as the codewords representing them. Note that the encoding map a→x is then simply x = aG That is, right multiplication by the generating matrix — a linear map!

25. Definition. An q-ary (n, M, d)-code is optimal if M = Aq (n, d). xn (deleting the k-th digit). This also acts, by restriction, on any subset of S n , and hence on any code C ∈ P (S n ), to produce a new code πk (C) ∈ P (S n−1). xn (replacing the k-th digit by i). Note that if D ∈ (n, M, d)-cod with d > 1 then |πk (D)| = M, since the maximum reduction in distance between distinct points caused by deleting one letter is 1 (so distinct points are still distinct after projection). 4. 26. Suppose d odd.

A 2-ary (n, M, d)-code exists iff a 2-ary (n + 1, M, d + 1)-code exists. Proof: (i) (Only if part): Let C ∈ (n, M, d)-cod. We construct C ′ ∈ (n + 1, M, d′)-cod (some d′ ) as follows. For each x ∈ C let x′ = x0 if w(x) even and x′ = x1 if w(x) odd. Note that d ≤ d′ ≤ d + 1. 24. Hence d′ = d + 1. (ii) (If part): Let D ∈ (n + 1, M, d + 1)-cod2 . Take x, y ∈ D such that d(x, y) = d + 1. Find a digit, the k-th say, where they differ. Construct D ′ ∈ (n, M, d′ )-cod2 by D ′ = πk (D). Note that d ≤ d′ ≤ d + 1.