Equivalence of codes

A general code of length $𝑛$ over alphabet $𝑆$ may be viewed as a subset of the function space $𝑆 Ω$ , where $| Ω | = 𝑛$ . Two codes $C \subseteq 𝑆 Ω$ and $D \subseteq 𝑇 Θ$ are equivalent iff there exist bijections $𝛼 : 𝑆 \to 𝑇$ and $𝜅 : Θ \to Ω$ such that $𝛼 𝜅 (C) = D$ code i.e. $𝜑$ is a bijection in the following commutative diagram in \Set:

When codes (and their alphabets) are given additional algebraic structure, we usually require a kind of equivalence which respects this structure. Examples include

Linear equivalence of codes, which is not strictly equivalence as $𝛼$ may vary in each digit

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Equivalence of codes

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