20 research outputs found
Several classes of 0-APN power functions over
Recently, the investigation of Partially APN functions has attracted a lot of
attention. In this paper, with the help of resultant elimination and MAGMA, we
propose several new infinite classes of 0-APN power functions over
. By the main result in [4], these -APN power functions
are CCZ-inequivalent to the known ones. Moreover, these infinite classes of
0-APN power functions can explain some exponents for which are
not yet ``explained" in the tables of Budaghyan et al. [3]
On the parameters of extended primitive cyclic codes and the related designs
Very recently, Heng et al. studied a family of extended primitive cyclic
codes. It was shown that the supports of all codewords with any fixed nonzero
Hamming weight of this code supporting 2-designs. In this paper, we study this
family of extended primitive cyclic codes in more details. The weight
distribution is determined. The parameters of the related -designs are also
given. Moreover, we prove that the codewords with minimum Hamming weight
supporting 3-designs, which gives an affirmative solution to Heng's conjecture
On correlation distribution of Niho-type decimation
The cross-correlation problem is a classic problem in sequence design. In
this paper we compute the cross-correlation distribution of the Niho-type
decimation over for any prime .
Previously this problem was solved by Xia et al. only for and in a
series of papers. The main difficulty of this problem for , as pointed
out by Xia et al., is to count the number of codewords of "pure weight" 5 in
-ary Zetterberg codes. It turns out this counting problem can be transformed
by the MacWilliams identity into counting codewords of weight at most 5 in
-ary Melas codes, the most difficult of which is related to a K3 surface
well studied in the literature and can be computed. When , the theory
of elliptic curves over finite fields also plays an important role in the
resolution of this problem
Codes and Pseudo-Geometric Designs from the Ternary -Sequences with Welch-type decimation
Pseudo-geometric designs are combinatorial designs which share the same
parameters as a finite geometry design, but which are not isomorphic to that
design. As far as we know, many pseudo-geometric designs have been constructed
by the methods of finite geometries and combinatorics. However, none of
pseudo-geometric designs with the parameters is constructed by the approach of coding theory. In
this paper, we use cyclic codes to construct pseudo-geometric designs. We
firstly present a family of ternary cyclic codes from the -sequences with
Welch-type decimation , and obtain some infinite family
of 2-designs and a family of Steiner systems
using these cyclic codes and their duals. Moreover, the parameters of these
cyclic codes and their shortened codes are also determined. Some of those
ternary codes are optimal or almost optimal. Finally, we show that one of these
obtained Steiner systems is inequivalent to the point-line design of the
projective space and thus is a pseudo-geometric design.Comment: 15 pages. arXiv admin note: text overlap with arXiv:2206.15153,
arXiv:2110.0388