A (v,Γ,λ) difference graph in an additive group G of order v is a copy of an abstract simple graph Γ with vertices in G such that the list of all possible differences between adjacent vertices covers ...every non-zero element of G exactly λ times. It is proved here that a (v,Γ,λ) difference graph cannot exist if v≡2(mod4) and Γ is Eulerian of odd size. As a consequence, we are able to exhibit infinitely many counterexamples to a recent conjecture proposed M. Buratti, A. Nakic and A. Wassermann.
There are two construction methods of designs from <inline-formula> <tex-math notation="LaTeX">(n,m) </tex-math></inline-formula>-bent functions, known as translation and addition designs. In this ...article we analyze, which equivalence relation for Boolean bent functions, i.e. <inline-formula> <tex-math notation="LaTeX">(n,1) </tex-math></inline-formula>-bent functions, and vectorial bent functions, i.e. <inline-formula> <tex-math notation="LaTeX">(n,m) </tex-math></inline-formula>-bent functions with <inline-formula> <tex-math notation="LaTeX">2\le m\le n/2 </tex-math></inline-formula>, is coarser: extended-affine equivalence or isomorphism of associated translation and addition designs. First, we observe that similar to the Boolean bent functions, extended-affine equivalence of vectorial <inline-formula> <tex-math notation="LaTeX">(n,m) </tex-math></inline-formula>-bent functions and isomorphism of addition designs are the same concepts for all even <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">m\le n/2 </tex-math></inline-formula>. Further, we show that extended-affine inequivalent Boolean bent functions in <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> variables, whose translation designs are isomorphic, exist for all <inline-formula> <tex-math notation="LaTeX">n\ge 6 </tex-math></inline-formula>. This implies, that isomorphism of translation designs for Boolean bent functions is a coarser equivalence relation than extended-affine equivalence. However, we do not observe the same phenomenon for vectorial bent functions in a small number of variables. We classify and enumerate all vectorial bent functions in six variables and show, that in contrast to the Boolean case, one cannot exhibit isomorphic translation designs from extended-affine inequivalent vectorial <inline-formula> <tex-math notation="LaTeX">(6,m) </tex-math></inline-formula>-bent functions with <inline-formula> <tex-math notation="LaTeX">m\in \{ 2,3 \} </tex-math></inline-formula>.
This study investigated the problem of constructing almost difference sets from single and unions of cyclotomic classes of order 10 (with and without zero) of the finite field GF(q), where q is a ...prime of the form q = 10n+1 for integer n≥1 and q<1000. Using exhaustive computer searches in Python, the method computed unions of two classes up to nine cyclotomic classes. Moreover, the equivalence of the generated almost difference sets having the same parameters was determined up to complementation. Results showed that a single cyclotomic class formed an almost difference set for q = 31, 71, and 151, while including zero produced an almost difference set for q=11, 31, 71, and 151. Additionally, Paley partial difference sets were constructed from the union of five cyclotomic classes. These findings addressed gaps in the literature on cyclotomy of order 10.
Partial difference sets in C2n×C2n Malandro, Martin E.; Smith, Ken W.
Discrete mathematics,
April 2020, Letnik:
343, Številka:
4
Journal Article
Recenzirano
Let Gn denote the group C2n×C2n, where Ck is the cyclic group of order k. We give an algorithm for enumerating the regular nontrivial partial difference sets (PDS) in Gn. We use our algorithm to ...obtain all of these PDS in Gn for 2≤n≤9, and we obtain partial results for n=10 and n=11. Most of these PDS are new. For n≤4 we also identify group-inequivalent PDS. Our approach involves constructing tree diagrams and canonical colorings of these diagrams. Both the total number and the number of group-inequivalent PDS in Gn appear to grow super-exponentially in n. For n=9, a typical canonical coloring represents in excess of 10146 group-inequivalent PDS, and there are precisely 2520 reversible Hadamard difference sets.
Almost Difference Sets have extensive applications in coding theory and cryptography. In this study, we introduce new constructions of Almost Difference Sets derived from cyclotomic classes of order ...12 in the finite field GF(q), where q is a prime satisfying the form q=12n+1 for positive integers n ≥ 1 and q < 1000. We show that a single cyclotomic class of order 12 (with and without zero) can form an almost difference set. Additionally, we successfully construct almost difference sets using unions of cyclotomic classes of order 12, both for even and odd values of n. To accomplish this, an exhaustive computer search employing Python was conducted. The method involved computing unions of two cyclotomic classes up to eleven classes and assessing the presence of almost difference sets. Finally, we classify the resulting almost difference sets with the same parameters up to equivalence and complementation.
This paper concentrates on a class of pseudorandom sequences generated by combining q-ary m-sequences and quadratic characters over a finite field of odd order, called binary generalized NTU ...sequences. It is shown that the relationship among the sub-sequences of binary generalized NTU sequences can be formulated as combinatorial structures called Hadamard designs. As a consequence, the combinatorial structures generalize the group structure discovered by Kodera et al. (IEICE Trans. Fundamentals, vol.E102-A, no.12, pp.1659-1667, 2019) and lead to a finite-geometric explanation for the investigated group structure.
This paper concentrates on a class of pseudorandom sequences generated by combining q-ary m-sequences and quadratic characters over a finite field of odd order, called binary generalized NTU ...sequences. It is shown that the relationship among the sub-sequences of binary generalized NTU sequences can be formulated as combinatorial structures called Hadamard designs. As a consequence, the combinatorial structures generalize the group structure discovered by Kodera et al. (IEICE Trans. Fundamentals, vol.E102-A, no.12, pp.1659-1667, 2019) and lead to a finite-geometric explanation for the investigated group structure.
Let be a prime of the form = + for integers ≥ and > . For < , we show that difference sets in the additive group of the field ( ) are obtained from unions of cyclotomic classes of orders = , , and ...and determine all such unions using a computer search. We then determine if the difference sets are equivalent to known cyclotomic or modified cyclotomic quadratic, quartic, sextic, or octic difference sets or their complements. This fills the gaps in the literature on the existence of difference sets from unions of cyclotomic classes for the specified orders. In addition, we extend Baumert and Fredricksen’s 1967 work on the construction of all inequivalent ( , , )-difference sets from unions of - cyclotomic classes of ( ) by constructing six inequivalent ( , , )-difference sets with zero added from unions of cyclotomic classes of order = .
Bent partitions Anbar, Nurdagül; Meidl, Wilfried
Designs, codes, and cryptography,
2022/4, Letnik:
90, Številka:
4
Journal Article
Recenzirano
Odprti dostop
Spread and partial spread constructions are the most powerful bent function constructions. A large variety of bent functions from a 2
m
-dimensional vector space
V
2
m
(
p
)
over
F
p
into
F
p
can be ...generated, which are constant on the sets of a partition of
V
2
m
(
p
)
obtained with the subspaces of the (partial) spread. Moreover, from spreads one obtains not only bent functions between elementary abelian groups, but bent functions from
V
2
m
(
p
)
to
B
, where
B
can be any abelian group of order
p
k
,
k
≤
m
. As recently shown (Meidl, Pirsic 2021), partitions from spreads are not the only partitions of
V
2
m
(
2
)
, with these remarkable properties. In this article we present first such partitions—other than (partial) spreads—which we call bent partitions, for
V
2
m
(
p
)
,
p
odd. We investigate general properties of bent partitions, like number and cardinality of the subsets of the partition. We show that with bent partitions we can construct bent functions from
V
2
m
(
p
)
into a cyclic group
Z
p
k
. With these results, we obtain the first constructions of bent functions from
V
2
m
(
p
)
into
Z
p
k
,
p
odd, which provably do not come from (partial) spreads.
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