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Thus, a vector v can be considered as a mapping from this set n to F , with vi the image of i ∈ n under v, v : n → F : i → vi . In this sense, the vector space can be identified with a set of mappings: F n = {v | v : n → F} . Of course, we also use the natural number n as the cardinality of sets of this order, but it should be always clear from the context if n means a set or a cardinality of a set. The metric principle which we are going to describe is based on the following fact: Definition and Theorem (Hamming metric) The function d : F n × F n → N : (u, v) → |{i | i ∈ n, ui = vi }| is a metric on the vector space F n , the Hamming metric.

A glance at Fig. 2 shows that the packing radius is the greatest integer which is strictly less than half the value of the minimum distance. c y ? ≥ d ≥ 2t + 1 ? c t Fig. 6 Corollary The packing radius of a linear code C is (dist(C ) − 1)/2 . 2 Linear Codes, Encoding and Decoding 15 Maximum-likelihood-decoding It is possible to correct up to t := (dist(C ) − 1)/2 errors in the following way (cf. Fig. 2): Using maximum-likelihood-decoding, a vector y ∈ F n is decoded into a codeword c ∈ C which is closest to y with respect to the Hamming metric.

The proof is based on the following fact which is immediate from the definition of group actions and which is of fundamental importance: gx = x ⇐⇒ x = g−1 x . The equivalence class G ( x ) = { gx | g ∈ G } of x ∈ X is called the G-orbit or, briefly, the orbit of x. We use the notation G \\ X := { G ( x ) | x ∈ X } to denote the set of orbits of G on X. A minimal but complete set T of orbit representatives is called a transversal of the orbits. 5 34 1. e. 6 ˙ ω = ˙ G ( t ). 7. It is easy to check that for any group action G X the orbits G ( x ) and G( x ), x ∈ X, coincide, whence, G \\ X = G\\ X.

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A Cellular Genetic Algorithm with Disturbances: Optimisation Using Dynamic Spatial Interactions by Kirley M.

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