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The generator polynomial of $\bar{C}$ is $‎\overline{g(x)}‎$. Because the conjugation operation is distributive over summation and multiplication.

Since my comment is long, I write it as an answer, but it is not a complete answer and just give some insight. Firstly, based on the paper you mentioned and based on the applications of MDS matrices …

When we study the structure of simple graphs with a lot of $1$ or $-1$ as its adjacency eigenvalues, the rank of its adjacency matrix is very important. The reason is, in these case, we can study the …

It is just a point of view. But it is more long for writing it as a comment. If $A$ be the adjacency matrix of graph $G$, then $R-A$ is its Laplacian matrix and there are some good bounds for the rad …

I think this book is suitable for further study about this question: "Representation Theory of Finite Groups An Introductory Approach" "Benjamin Steinberg" This book has some chapters related to you …

There are some good classification of Integral Cayley graphs, which by your terminology means all their adjacency matrix eigenvalues are integer. The adjacency matrices of Cayley graphs over cyclic gr …

There are a lot of work in this direction. For an updated (and also very interesting) book, you can see: "inequalities for graph eigenvalues" by Zoran Stanić. Especially, you can see the chapter two …

This is an optimistic approach and since it is long, I write it as an answer. First we set $$H(M,x)=\Sigma_{x\in \mathbb{Z}^n}{e^{-x^TMx}},$$ Which is your introduced summation. Let $J$ be the all one …

Maybe these books be interesting: Linear Algebra for Quantum Theory Per-Olov Löwdin Quantum Algorithms via Linear Algebra: A Primer Richard J. Lipton Kenneth W. Regan Quantum Computing: From Lin …

It is not a complete solution, but can be such of them by some calculations. You can assign a graph to the matrice $M$ in each case and analyze them. If all-zero blocks have equal size, their size $t …