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In coding theory, the notions of coding gain and minimum determinant of a code have been defined as follows: let $\mathcal{X}$ be a (full diversity) code and $X,X^\prime\in\mathcal{X}$.

Then the $\textit{coding gain}$ is $\operatorname{det}\left(\left(X-X^{\prime}\right)\left(X-X^{\prime}\right)^{\dagger}\right)$, and the $\textit{minimum determinant}$ is $min_{X\ne X^\prime\in\mathcal{X}}\operatorname{det}\left(\left(X-X^{\prime}\right)\left(X-X^{\prime}\right)^{\dagger}\right)$.

Do these notions find application in cryptographic terms in code-based cryptography? That is, do properties of the minimum determinant have consequences to a cryptographer building a cryptosystem, or in cryptography in general?

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    $\begingroup$ Write down the explicit definition of coding gain please. And the explicit minimisation for the other. What are you minimising over? $\endgroup$
    – kodlu
    Commented Jul 22, 2021 at 19:47
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    $\begingroup$ I've added the definitions. $\endgroup$
    – a196884
    Commented Jul 23, 2021 at 10:04

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I don't believe so. The two concepts are related to fading channels with continuous noise and how fast certain iterative decoding algorithms converge. I cannot think of a relevance to code based cryptography.

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