# Coding gain and minimum determinant in cryptography

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?

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