17,538 research outputs found
Effective Construction of a Class of Bent Quadratic Boolean Functions
In this paper, we consider the characterization of the bentness of quadratic
Boolean functions of the form where ,
is even and . For a general , it is difficult to determine
the bentness of these functions. We present the bentness of quadratic Boolean
function for two cases: and , where and are two
distinct primes. Further, we give the enumeration of quadratic bent functions
for the case
A Note on Cyclic Codes from APN Functions
Cyclic codes, as linear block error-correcting codes in coding theory, play a
vital role and have wide applications. Ding in \cite{D} constructed a number of
classes of cyclic codes from almost perfect nonlinear (APN) functions and
planar functions over finite fields and presented ten open problems on cyclic
codes from highly nonlinear functions. In this paper, we consider two open
problems involving the inverse APN functions and the Dobbertin
APN function . From the calculation of
linear spans and the minimal polynomials of two sequences generated by these
two classes of APN functions, the dimensions of the corresponding cyclic codes
are determined and lower bounds on the minimum weight of these cyclic codes are
presented. Actually, we present a framework for the minimal polynomial and
linear span of the sequence defined by ,
where is a primitive element in . These techniques can also be
applied into other open problems in \cite{D}
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