By Paul P. Martin

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**Extra info for Bottom up Computing and Discrete Mathematics**

**Example text**

1) In f3 we send 01101. 1). g. 0009801496. 5 times reduced error probability 20-fold! 4 and above it is better than f3 ). Anyway, the point is it makes a difference. So the science of coding theory is non-trivial. The game is ON! 14. Definition. A q-ary (n, M, d)-code is a block length n code with M codewords and minimum distance d. 4. OPTIMISATION For S a set let P (S) denote the power set of S. Thus P (S n ) is the set of length-n |S|-ary codes; and a q-ary (n, M, d)-code C is an element of P (S n ) (some S of degree q) such that |C| = M and d(C) = d.

A reflexive, symmetric, transitive relation. 62. Let C be a linear code generated by G. 6. LINEAR CODES (C1) permuting columns (C2) multiplying a column by a non-zero scalar a ∈ Fq . Then G′ generates C ′ an equivalent linear code to C. ). A 21 22 2,n−k G′ = = [1k |A] ... Ak,n−k where 1k is the k × k unit matrix and A has entries in Fq . 63. Example. 64. Exercise. Let Ci be the 3-ary code generated by Gi , where G1 = 1011 0112 , G2 = 1011 0111 For each of i = 1, 2, list Ci and hence compute d(Ci ).

To make a d = 1 code, then, all we have to do is make any code at all. The biggest q-ary length n code has M = q n (just include every possible codeword). For binary n = 3, therefore, this biggest code has M = 8. That is, for (3,8,1): {000, 001, 010, 011, 100, 101, 110, 111} is the unique such code. 4. OPTIMISATION 49 increase the distance by 1 from our (3,8,1) code: {0000, 0011, 0101, 0110, 1001, 1010, 1100, 1111} For our fourth case it is no longer obvious how to construct a code. Under the circumstances it is prudent to check if such a code is impossible, by checking the BP and singleton bounds.