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On the dimension of twisted centralizer codes. (English) Zbl 1398.94243

Summary: Given a field \(F\), a scalar \(\lambda \in F\) and a matrix \(A \in F^{n \times n}\), the twisted centralizer code \(C_F(A, \lambda) : = \{B \in F^{n \times n} | A B - \lambda B A = 0 \}\) is a linear code of length \(n^2\) over \(F\). When \(A\) is cyclic and \(\lambda \neq 0\) we prove that \(\dim C_F(A, \lambda) = \deg(\gcd(c_A(t), \lambda^n c_A(\lambda^{- 1} t)))\) 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 \(\dim C_F(A, \lambda) \le n^2 / 2\) for \(\lambda \notin \{0, 1 \}\) and ’almost all’ \(n \times n\) matrices \(A\) over \(F\).

MSC:

94B05 Linear codes (general theory)
94B65 Bounds on codes
60C05 Combinatorial probability
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References:

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