On the concept of code-isomorphy

Abstract

LetF be a finite set of cardinality ¦F¦ =q ≥2,n ≥ 1 an integer and ϱ:F ⁿ×Fⁿ→ℕ₀ theHamming metric. Acode isomorphism C → D between two block codesC,D $\subseteq$ Fⁿ is defined as an isometry which can be extended to an isometry of the whole space Fⁿ. Any permutation π ∈S _n of the positions canonically induces a so-calledequivalence map $\tilde \pi$ ∈ Aut Fⁿ; any system κ ≔ (κ₁,κ₂,...,κ_n) ofn permutations of the character setF induces a so-calledconfiguration $\tilde \kappa$ ∈ Aat Fⁿ. The group Aut Fⁿ of all isometries of Fⁿ turns out to be semidirect product of the configuration group with the symmetric group of degreen. The codeword estimating failure probability of a maximum likelihood codeword estimator for aq-nary symmetric channel does not depend on the transmitted codeword, if the automorphism group of the code acts transitively on the set of codewords. When using a systematic (n, k)-encoder, the symbol decoding failure probability does not depend on the transmitted symbol or on the time of transmission if the configuration group and the automorphism group act transitively on the set of codewords resp. on the set of thek information positions.