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By a result of Latimer and MacDuffee, there are a finite number of equivalence classes of n × n matrices over F q [ T ] with minimum polynomial p (X) , where p is an n th degree polynomial, irreducible over F q [ T ]. In this paper, we develop an algorithm for finding a canonical representative of each matrix class, for p (X) = X 2 − Γ X − Δ ∈ F q [ T ] [ X ]. [ABSTRACT FROM AUTHOR]

In this paper, we provide two distinct list decoding algorithms tailored for specific classes of maximal order number field codes. The first algorithm focuses on the unit codes, while the second algorithm is dedicated to the Arakelov divisor-based number field codes. These codes were originally introduced by Maire and Oggier. To achieve our decoding goals, we utilize the list decoding algorithm for general number field codes proposed by Biasse and Quintin. Our approach involves carefully calculating the required parameter B from Biasse and Quintin's algorithm, and adjusting the returned list to satisfy the desired conditions. For unit codes, we consider two cases: when the underlying number field is totally real and when it has an arbitrary signature. We separate these cases because the parameter B can be improved in the totally real scenario. Additionally, we present our list decoding algorithm for Arakelov divisor-based number field codes, focusing on number fields with arbitrary signatures. [ABSTRACT FROM AUTHOR]