An asymptotic property of quaternary additive codes

Let $n_k(s)$ be the maximal length $n$ such that a quaternary additive $[n,k,n-s]_4$-code exists. We solve a natural asymptotic problem by determining the lim sup $\lambda_k$ of $n_k(s)/s,$ and the smallest value of $s$ such that $n_k(s)/s=\lambda_k.$ Our new family of quaternary additive codes has parameters $[4^k-1,k,4^k-4^{k-1}]_4=[2^{2k}-1,k,3\cdot 2^{2k-2}]_4$ (where $k=l/2$ and $l$ is an odd integer). These are constant-weight codes. The binary codes obtained by concatenation meet the Griesmer bound with equality. The proof is in terms of multisets of lines in $PG(l-1,2).$Comment: 8 page

New semifields, PN and APN functions

We describe a method of proving that certain functions f : F → F defined on a finite field F are either PN-functions (in odd characteristic) or APN-functions (in characteristic 2). This method is illustrated by giving short proofs of the APN-respectively the PN-property for various families of functions. The main new contribution is the construction of a family of PN-functions and their corresponding commutative semifields of dimension 4s in arbitrary odd characteristic. It is shown that a subfamily of order p 4s for odd s \u3e 1 is not isotopic to previously known examples. © 2009 Springer Science+Business Media, LLC

An infinite family of 7-designs

We study designs in the binary affine space invariant under the affine group in its 3-transitive action. The main result is a family 7-(2n,8,45), n≥6 of non-simple designs. We also obtain 5-(2n,6,3) for every n≥3 and 5-(2n,7,7(2n - 16)/2) for every even n≥6. © 2001 Elsevier Science B.V. All rights reserved

A family of crooked functions

It has been proved in Bierbrauer and Kyureghyan (Des. Codes Cryptogr. 46:269-301, 2008) that a binomial function aX i + bX j can be crooked only if both exponents i, j have 2-weight ≤ 2. In the present paper we give a brief construction for all known examples of crooked binomial functions. These consist of an infinite family and one sporadic example. The construction of the sporadic example uses the properties of an algebraic curve of genus 3. Computer experiments support the conjecture that each crooked binomial is equivalent either to a member of the family or to the sporadic example. © 2008 Springer Science+Business Media, LLC

A2-codes from universal hash classes

We describe a general method to construct codes for unconditional authentication with arbitration (A2-codes), which protect not only against outside opponents but also against certain types of frauds from the receiver and transmitter. The constructions are based on orthogonal arrays and universal hash classes. The idea is to construct A2-codes out of pairs of A-codes. The hitherto known examples are special cases of the construction. Along the way we also construct new universal hash classes

Nordstrom-Robinson code and A \u3c inf\u3e 7 -geometry

The Nordstrom-Robinson code NR is a nonlinear binary code of length 16, with 28 codewords and minimum distance 6. Its automorphism group is a semidirect product of an elementary abelian group of order 16 and the alternating group A7 . This group and the corresponding action of A7 is also at the origin of the sporadic A7-geometry. We construct this geometry and derive the Nordstrom-Robinson code from it. © 2006 Elsevier Inc. All rights reserved