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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!

Adjoining a root of f to Z2 we get a number system consisting of {0, 1, x, 1 + x}, and that’s it! p) This field is called F4 . More generally we can adjoin a root of an irreducible polynomial of degree e and get F2e . More generally still, Fpe . e. it is interested in the minimum distance d(C) and so on). From this point of view, the precise choice of symbols used in codewords is not directly relevant. However, realistically, the message itself is quite likely to take the form of strings of letter from some human alphabet — and the recovery of the correct letters at the end of the process is the essential aim.

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.

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