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In [6], Lilya Budaghyan and Claude Carlet introduced a family of APN functions on F"2"^"2"^"k of the form F(x)=x(x^2^^^i+x^2^^^k+cx^2^^^k^^^+^^^i)+x^2^^^i(c^2^^^kx^2^^^[email protected]^2^^^k^^^+^^^i)+x^2^^^k^^^+^^^i^+^2^^^k. They showed that this infinite family exists provided the existence of the quadratic polynomial G(y)=y^2^^^i^+^1+cy^2^^^i+c^2^^^ky+1, which has no zeros such that y^2^^^k^+^1=1, or in particular has no zeros in F"2"^"2"^"k. However, up to now, no construction of such polynomials is known. In this paper, we show that, when k is an odd integer, the APN function F is CCZ-equivalent to the one in [2, Theorem 1]; and when k is even with [email protected]?k, we explicitly construct the polynomial G, and hence demonstrate the existence of F. More generally, it is well known that G relates to the polynomial P"a(x)=x^2^^^i^+^[email protected]?F"2"^"n[x] and P"a has applications in many other contexts. We determine all coefficients a such that P"a has no zeros on F"2"^"n when gcd(i,n)=1 and n is even.

In this paper, we give the first example of a non-cyclic triple-error-correcting code which is not equivalent to the primitive BCH code. It has parameters [63, 45, 7]. We also give better bounds on minimum distances of some [2 n ? 1, 2 n - 3n - 1] cyclic codes with three small zeroes. Finally, we reprove weight distribution results of Kasami for triple-error-correcting BCH-like codes using direct methods.

We define a pseudo quasi-3 design as a symmetric design with the property that the derived and residual designs with respect to at least one block are quasi-symmetric. Quasi-symmetric designs can be used to construct optimal self complementary codes. In this article we give a construction of an infinite family of pseudo quasi-3 designs whose residual designs allow us to construct a family of codes with a new parameter set that meet the Grey Rankin bound., Comment: 9 pages, submitted to the European Journal of Comniatorics

We introduce two new infinite families of APN functions, one on fields of order 22k for k not divisible by 2, and the other on fields of order 23k for k not divisible by 3. The polynomials in the first family have between three and k+2 terms, the second family's polynomials have three terms.

In the early 1980s Dan Hughes wrote two papers on semi-symmetric 3-designs, which are closely related to quasi-3 designs. In 1973 Peter Cameron asked whether the dual design of any quasi-3 design must also be quasi-3. In this article, we answer that question. We give some general constructions of quasi-3 designs which enable us to give the answer. The answer is no.

We present a new technique for constructing binary error correcting codes and give some examples of codes that can be constructed via this method. Among the examples is an infinite family of self-complementary codes with parameters (2u 2?u, 8u 2, u 2?u) that can be constructed whenever there exists a u × u Hadamard Matrix. These codes meet the Grey---Rankin bound and imply the existence of quasi-symmetric designs on 2u 2?u points.

Using Hadamard matrices and mutually orthogonal Latin squares, we construct two new quasi-symmetric designs, with parameters 2 ? (66,30,29) and 2 ? (78,36,30). These are the first examples of quasi-symmetric designs with these parameters. The parameters belong to the families 2 ? (2u 2 ? u,u 2 ? u,u 2 ? u ? 1) and 2 ? (2u 2 + u,u 2,u 2 ? u), which are related to Hadamard parameters. The designs correspond to new codes meeting the Grey---Rankin bound.

A theorem of Bannai and Sawano shows that certain four-weight spin models exist if and only if certain quasi-3 designs exist. We verify that the known quasi-3 designs, other than SDP designs, do not give rise to four-weight spin models.

In this article we construct an infinite family of linear error correcting codes over F q for any prime power q. The code parameters are [q2t + qt−1 − q2t−1 − qt, 2t+1, q2t + q2t−2 + qt−1 − 2q2t−1 − qt] q , for any positive integer t. This family is a generalisation of the optimal self-complementary binary codes with parameters [2u2 − u, 2t + 1, u2 − u]2, where u = 2t−1. The codes are obtained by considering a submatrix of a specially constructed generalised Hadamard matrix. The optimality of the family is confirmed by using a recently derived generalisation of the Grey-Rankin bound when t > 1, and the Griesmer bound when t = 1.

Establishing the CCZ-equivalence of a pair of APN functions is generally quite difficult. In some cases, when seeking to show that a putative new infinite family of APN functions is CCZ inequivalent to an already known family, we rely on computer calculation for small values of n. In this paper we present a method to prove the inequivalence of quadratic APN functions with the Gold functions. Our main result is that a quadratic function is CCZ-equivalent to an APN Gold function if and only if it is EA-equivalent to that Gold function. As an application of this result, we prove that a trinomial family of APN functions that exist on finite fields of order 2^n where n = 2 mod 4 are CCZ inequivalent to the Gold functions. The proof relies on some knowledge of the automorphism group of a code associated with such a function., 13 pg

Functions with low differential uniformity can be used as the s-boxes of symmetric cryptosystems as they have good resistance to differential attacks. The AES (Advanced Encryption Standard) uses a differentially-4 uniform function called the inverse function. Any function used in a symmetric cryptosystem should be a permutation. Also, it is required that the function is highly nonlinear so that it is resistant to Matsui's linear attack. In this article we demonstrate that a highly nonlinear permutation discovered by Hans Dobbertin has differential uniformity of four and hence, with respect to differential and linear cryptanalysis, is just as suitable for use in a symmetric cryptosystem as the inverse function., Comment: 10 pages, submitted to Finite Fields and Their Applications

We present two infinite families of APN functions where the degree of the field is divisible by 3 but not 9. Our families contain two already known families as special cases. We also discuss the inequivalence proof (by computation) which shows that these functions are new., 12 pages, submitted to Cryptography and Communications

In this paper we compute the Fourier spectra of some recently discovered binomial APN functions. One consequence of this is the determination of the nonlinearity of the functions, which measures their resistance to linear cryptanalysis. Another consequence is that certain error-correcting codes related to these functions have the same weight distribution as the 2-error-correcting BCH code. Furthermore, for fields of odd degree, our results provide an alternative proof of the APN property of the functions., 20 pages. Submitted to the SIAM Journal on Discrete Mathematics

It is well known that a quadratic function defined on a finite field of odd degree is almost bent (AB) if and only if it is almost perfect nonlinear (APN). For the even degree case there is no apparent relationship between the values in the Fourier spectrum of a function and the APN property. In this article we compute the Fourier spectrum of the new quadranomial family of APN functions. With this result, all known infinite families of APN functions now have their Fourier spectra and hence their nonlinearities computed., Comment: 12 pages, submitted to Adavances in the Mathematics of communication

We compute the Walsh spectrum and hence the nonlinearity of a new family of quadratic multi-term APN functions. We show that the distribution of values in the Walsh spectrum of these functions is the same as the Gold function.

We characterize the SDP designs that give rise to four-weight spin models with two values. We prove that the only such designs are the symplectic SDP designs. The proof involves analysis of cardinalities of intersections of four blocks., 11 pages

Differentially 4 uniform permutations with high nonlinearity on fields of even degree are crucial to the design of S-boxes in many symmetric cryptographic algorithms. Until now, there are not many known such functions and all functions known are power functions. In this paper, we construct the first class of binomial differentially 4 uniform permutations with high nonlinearity on F 2 6 m , where m is an odd integer. This result gives a positive answer to an open problem proposed in Bracken and Leander (2010) [7] .