A major contribution of [1] is a reduction of the problem of correcting errors in quantum computations to the construction of codes in binary symplectic spaces. This mechanism is known as the additive or stabilizer construction. We consider an obvious generalization of these quantum codes in the symplectic geometry setting and obtain general constructions using our theory of twisted BCH‐codes (also known as Reed–Solomon subspace subcodes). This leads to families of quantum codes with good parameters. Moreover, the generator matrices of these codes can be described in a canonical way. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 174–188, 2000

Following an example in [12], we show how to change one coordinate function of an almost perfect nonlinear (APN) function in order to obtain new examples. It turns out that this is a very powerful method to construct new APN functions. In particular, we show that our approach can be used to construct a "non-quadratic" APN function. This new example is in remarkable contrast to all recently constructed functions which have all been quadratic. An equivalent function has been found independently by Brinkmann and Leander [8]. However, they claimed that their function is CCZ equivalent to a quadratic one. In this paper we give several reasons why this new function is not equivalent to a quadratic one

For a finite abelian group G, let s(G) denote the smallest integer l such that every sequence S over G of length |S| ≥ l has a zero-sum subsequence of length exp(G). We derive new upper and lower bounds for s(G), and all our bounds are sharp for special types of groups. The results are not restricted to groups G of the form G = C r n , but they respect the structure of the group. In particular, we show s(C 4 n) ≥ 20n − 19 for all odd n, which is sharp if n is a power of 3. Moreover, we investigate the relationship between extremal sequences and maximal caps in finite geometry.

A new almost perfect nonlinear function (APN) on F 2 10 which is not equivalent to any of the previously known APN mappings is constructed. This is the first example of an APN mapping which is not equivalent to a power mapping.

We prove that 45 is the size of the largest caps in AGð5; 3Þ; and such a 45-cap is always obtained from the 56-cap in PGð5; 3Þ by deleting an 11-hyperplane. # 2002 Elsevier Science (USA)