28,009 research outputs found
Codes, graphs and schemes from nonlinear functions
AbstractWe consider functions on binary vector spaces which are far from linear functions in different senses. We compare three existing notions: almost perfect nonlinear functions, almost bent (AB) functions, and crooked (CR) functions. Such functions are of importance in cryptography because of their resistance to linear and differential attacks on certain cryptosystems. We give a new combinatorial characterization of AB functions in terms of the number of solutions to a certain system of equations, and a characterization of CR functions in terms of the Fourier transform. We also show how these functions can be used to construct several combinatorial structures; such as semi-biplanes, difference sets, distance regular graphs, symmetric association schemes, and uniformly packed (BCH and Preparata) codes
On diameter perfect constant-weight ternary codes
From cosets of binary Hamming codes we construct diameter perfect
constant-weight ternary codes with weight (where is the code length)
and distances 3 and 5. The class of distance 5 codes has parameters unknown
before. Keywords: constant-weight codes, ternary codes, perfect codes, diameter
perfect codes, perfect matchings, Preparata codesComment: 15 pages, 2 figures; presented at 2004 Com2MaC Conference on
Association Schemes, Codes and Designs; submitted to Discrete Mathematic
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}
- …