A characterization of linearized polynomials with maximum kernel

We provide sufficient and necessary conditions for the coefficients of a q-polynomial f over Fqn which ensure that the number of distinct roots of f in Fqn equals the degree of f. We say that these polynomials have maximum kernel. As an application we study in detail q-polynomials of degree qn−2 over Fqn which have maximum kernel and for n≤6 we list all q-polynomials with maximum kernel. We also obtain information on the splitting field of an arbitrary q-polynomial. Analogous results are proved for qs-polynomials as well, where gcd⁡(s,n)=1

A characterization of linearized polynomials with maximum kernel

We provide sufficient and necessary conditions for the coefficients of a q-polynomial f over GF(q^n) which ensure that the number of distinct roots of f in GF(q^n) equals the degree of f. We say that these polynomials have maximum kernel. As an application we study in detail q-polynomials of degree q^(n-2) over GF(q^n) which have maximum kernel and for n<7 we list all q-polynomials with maximum kernel. We also obtain information on the splitting field of an arbitrary q-polynomial. Analogous results are proved for q^s-polynomials as well, where gcd(s,n) = 1

On interpolation-based decoding of a class of maximum rank distance codes

In this paper we present an interpolation-based decoding algorithm to decode a family of maximum rank distance codes proposed recently by Trombetti and Zhou. We employ the properties of the Dickson matrix associated with a linearized polynomial with a given rank and the modified Berlekamp-Massey algorithm in decoding. When the rank of the error vector attains the unique decoding radius, the problem is converted to solving a quadratic polynomial, which ensures that the proposed decoding algorithm has polynomial-time complexity.acceptedVersio

Solving Xq+1+X+a=0X^{q+1}+X+a=0 over Finite Fields

Solving the equation $P_a(X):=X^{q+1}+X+a=0$ over finite field \GF{Q}, where $Q=p^n, q=p^k$ and $p$ is a prime, arises in many different contexts including finite geometry, the inverse Galois problem \cite{ACZ2000}, the construction of difference sets with Singer parameters \cite{DD2004}, determining cross-correlation between $m$-sequences \cite{DOBBERTIN2006,HELLESETH2008} and to construct error-correcting codes \cite{Bracken2009}, as well as to speed up the index calculus method for computing discrete logarithms on finite fields \cite{GGGZ2013,GGGZ2013+} and on algebraic curves \cite{M2014}. Subsequently, in \cite{Bluher2004,HK2008,HK2010,BTT2014,Bluher2016,KM2019,CMPZ2019,MS2019}, the \GF{Q}-zeros of $P_a(X)$ have been studied: in \cite{Bluher2004} it was shown that the possible values of the number of the zeros that $P_a(X)$ has in \GF{Q} is $0$, $1$, $2$ or $p^{\gcd(n, k)}+1$. Some criteria for the number of the \GF{Q}-zeros of $P_a(x)$ were found in \cite{HK2008,HK2010,BTT2014,KM2019,MS2019}. However, while the ultimate goal is to identify all the \GF{Q}-zeros, even in the case $p=2$, it was solved only under the condition $\gcd(n, k)=1$ \cite{KM2019}. We discuss this equation without any restriction on $p$ and $\gcd(n,k)$. New criteria for the number of the \GF{Q}-zeros of $P_a(x)$ are proved. For the cases of one or two \GF{Q}-zeros, we provide explicit expressions for these rational zeros in terms of $a$. For the case of $p^{\gcd(n, k)}+1$ rational zeros, we provide a parametrization of such $a$'s and express the $p^{\gcd(n, k)}+1$ rational zeros by using that parametrization