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By Paul P. Martin

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We might as well start with a quadratic. Since we want to end up with ‘coefficients’ in Z2 the coefficients in the polynomial need to be in Z2 . There is then only one irreducible polynomial available: f (x) = 1+x+x2 . 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).

A q-ary [n, k, d]-code is a linear code in Fqn of dim k and minimum distance d. Write [n, k, d] − cod for the set of all such (with q understood). 6. LINEAR CODES Thus C ∈ [n, k, d] − cod implies C ∈ (n, q k , d) − cod, but the converse is false. 54. Example. Our first three examples are all binary linear codes: C1 ∈ [2, 2, 1] − cod; C2 ∈ [3, 2, 2] − cod; C3 ∈ [5, 2, 3] − cod. Exercise: check this. Recall that for a general code we need 21 |C|(|C| − 1) calculations to compute d(C). We can radically reduce this for a linear code.

Consider the collection of t-balls centred on all x ∈ C. This is a ‘fuzzy picture’ of the elements x. 4. 4: Ball packing heuristic (using Euclidean metric). of uncertainty in it, in a neighbourhood of S, caused by up to t transmission errors. 4): (a) If d(C) ≥ t + 1 then no x lies in another’s ball. Thus if 1 up to t errors occur then the received message is not in C and we know we have an error. (b) If d(C) ≥ 2t + 1 then even the balls are disjoint (this is perhaps not so obvious with the Hamming distance, cf.