24,956 research outputs found
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}
On Equivalence of Known Families of APN Functions in Small Dimensions
In this extended abstract, we computationally check and list the
CCZ-inequivalent APN functions from infinite families on for n
from 6 to 11. These functions are selected with simplest coefficients from
CCZ-inequivalent classes. This work can simplify checking CCZ-equivalence
between any APN function and infinite APN families.Comment: This paper is already in "PROCEEDING OF THE 20TH CONFERENCE OF FRUCT
ASSOCIATION
On the Derivative Imbalance and Ambiguity of Functions
In 2007, Carlet and Ding introduced two parameters, denoted by and
, quantifying respectively the balancedness of general functions
between finite Abelian groups and the (global) balancedness of their
derivatives , (providing an
indicator of the nonlinearity of the functions). These authors studied the
properties and cryptographic significance of these two measures. They provided
for S-boxes inequalities relating the nonlinearity to ,
and obtained in particular an upper bound on the nonlinearity which unifies
Sidelnikov-Chabaud-Vaudenay's bound and the covering radius bound. At the
Workshop WCC 2009 and in its postproceedings in 2011, a further study of these
parameters was made; in particular, the first parameter was applied to the
functions where is affine, providing more nonlinearity parameters.
In 2010, motivated by the study of Costas arrays, two parameters called
ambiguity and deficiency were introduced by Panario \emph{et al.} for
permutations over finite Abelian groups to measure the injectivity and
surjectivity of the derivatives respectively. These authors also studied some
fundamental properties and cryptographic significance of these two measures.
Further studies followed without that the second pair of parameters be compared
to the first one.
In the present paper, we observe that ambiguity is the same parameter as
, up to additive and multiplicative constants (i.e. up to rescaling). We
make the necessary work of comparison and unification of the results on ,
respectively on ambiguity, which have been obtained in the five papers devoted
to these parameters. We generalize some known results to any Abelian groups and
we more importantly derive many new results on these parameters
On the Weight Distribution of Codes over Finite Rings
Let R > S be finite Frobenius rings for which there exists a trace map T from
R onto S as left S modules. Let C:= {x -> T(ax + bf(x)) : a,b in R}. Then C is
an S-linear subring-subcode of a left linear code over R. We consider functions
f for which the homogeneous weight distribution of C can be computed. In
particular, we give constructions of codes over integer modular rings and
commutative local Frobenius that have small spectra.Comment: 18 p
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