On the dimension of twisted centralizer codes

Abstract

Given a field $F$, a scalar $\lambda \in F$ and a matrix $A\in F^{n\times n}$, the twisted centralizer code $C_F(A,\lambda ):=\{B\in F^{n\times n}\mid AB-\lambda BA=0\}$ is a linear code of length $n^2$. When $A$ is cyclic and $\lambda \ne 0$ we prove that $\dim C_F(A,\lambda )=\mathrm{deg}(\gcd (c_A(t),{\lambda }^n{c}_A({\lambda }^{-1}t)))$ where $c_A(t)$ denotes the characteristic polynomial of $A$. We also show how $C_F(A,\lambda )$ decomposes, and we estimate the probability that $C_F(A,\lambda )$ is nonzero when $|F|$ is finite. Finally, we prove $\dim C_F(A,\lambda )⩽n^2/2$ for $\lambda \notin \{0,1\}$ and `almost all' matrices $A$.

17 pages, 2 figures Proof of Theorem 2.8 altered: last line and third last line changed

Cited by in corpus (1)

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