共查询到20条相似文献,搜索用时 46 毫秒
1.
John Polhill 《Designs, Codes and Cryptography》2008,46(3):365-377
A partial difference set having parameters (n
2, r(n − 1), n + r
2 − 3r, r
2 − r) is called a Latin square type partial difference set, while a partial difference set having parameters (n
2, r(n + 1), − n + r
2 + 3r, r
2 + r) is called a negative Latin square type partial difference set. Nearly all known constructions of negative Latin square partial difference sets are in elementary
abelian groups. In this paper, we develop three product theorems that construct negative Latin square type partial difference
sets and Latin square type partial difference sets in direct products of abelian groups G and G′ when these groups have certain Latin square or negative Latin square type partial difference sets. Using these product theorems,
we can construct negative Latin square type partial difference sets in groups of the form where the s
i
are nonnegative integers and s
0 + s
1 ≥ 1. Another significant corollary to these theorems are constructions of two infinite families of negative Latin square
type partial difference sets in 3-groups of the form for nonnegative integers s
i
. Several constructions of Latin square type PDSs are also given in p-groups for all primes p. We will then briefly indicate how some of these results relate to amorphic association schemes. In particular, we construct
amorphic association schemes with 4 classes using negative Latin square type graphs in many nonelementary abelian 2-groups;
we also use negative Latin square type graphs whose underlying sets can be elementary abelian 3-groups or nonelementary abelian
3-groups to form 3-class amorphic association schemes.
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2.
Relatively few constructions are known of negative Latin square type Partial Difference Sets (PDSs), and most of the known constructions are in elementary abelian groups. We present a product construction that produces negative Latin square type PDSs, and we apply this product construction to generate examples in p-groups of exponent bigger than p. 相似文献
3.
Xiang‐Dong Hou 《组合设计杂志》2002,10(6):394-402
Latin square type partial difference sets (PDS) are known to exist in R × R for various abelian p‐groups R and in ?t. We construct a family of Latin square type PDS in ?t × ?2ntp using finite commutative chain rings. When t is odd, the ambient group of the PDS is not covered by any previous construction. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 394–402, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10029 相似文献
4.
There exist few examples of negative Latin square type partial difference sets (NLST PDSs) in nonabelian groups. We present a list of 176 inequivalent NLST PDSs in 48 nonisomorphic, nonabelian groups of order 64. These NLST PDSs form 8 nonisomorphic strongly regular graphs. These PDSs were constructed using a combination of theoretical techniques and computer search, both of which are described. The search was run exhaustively on 212/267 nonisomorphic groups of order 64. 相似文献
5.
John Polhill 《Designs, Codes and Cryptography》2009,52(2):163-169
By modifying a construction for Hadamard (Menon) difference sets we construct two infinite families of negative Latin square
type partial difference sets in groups of the form where p is any odd prime. One of these families has the well-known Paley parameters, which had previously only been constructed in
p-groups. This provides new constructions of Hadamard matrices and implies the existence of many new strongly regular graphs
including some that are conference graphs. As a corollary, we are able to construct Paley–Hadamard difference sets of the
Stanton-Sprott family in groups of the form when is a prime power. These are new parameters for such difference sets.
相似文献
6.
Large sets of disjoint group‐divisible designs with block size three and type 2n41 were first studied by Schellenberg and Stinson because of their connection with perfect threshold schemes. It is known that such large sets can exist only for n ≡0 (mod 3) and do exist for all odd n ≡ (mod 3) and for even n=24m, where m odd ≥ 1. In this paper, we show that such large sets exist also for n=2k(3m), where m odd≥ 1 and k≥ 5. To accomplish this, we present two quadrupling constructions and two tripling constructions for a special large set called *LS(2n). © 2002 Wiley Periodicals, Inc. J Combin Designs 11: 24–35, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10032 相似文献
7.
Zeying Wang 《组合设计杂志》2020,28(2):149-152
Partial difference sets with parameters are called Paley type partial difference sets. In this note, we prove that if there exists a Paley type partial difference set in an abelian group of order v, where v is not a prime power, then or , an odd integer. In 2010, Polhill constructed Paley type partial difference sets in abelian groups with those orders. Thus, combining with the constructions of Polhill and the classical Paley construction using nonzero squares of a finite field, we completely answer the following question: “For which odd positive integers , can we find a Paley type partial difference set in an abelian group of order ?” 相似文献
8.
In this paper new lower bounds for the cardinality of minimal m-blocking sets are determined. Let r2(q) be the number such that q+r2(q)+1 is the cardinality of the smallest non-trivial line-blocking set in a plane of order q. If B is a minimal m-blocking set in PG(n,q) that contains at most qm+qm−1+…+q+1+r2(q)·(∑i=2m−n′m−1qi) points for an integer n′ satisfying mn′2m, then the dimension of B is at most n′. If the dimension of B is n′, then the following holds. The cardinality of B equals qm+qm−1+…+q+1+r2(q)(∑i=2m−n′m−1qi). For n′=m the set B is an m-dimensional subspace and for n′=m+1 the set B is a cone with an (m−2)-dimensional vertex over a non-trivial line-blocking set of cardinality q+r2(q)+1 in a plane skew to the vertex. This result is due to Heim (Mitt. Math. Semin. Giessen 226 (1996), 4–82). For n′>m+1 and q not a prime the number q is a square and for q16 the set B is a Baer cone. If q is odd and |B|<qm+qm−1+…+q+1+r2(q)(qm−1+qm−2), it follows from this result that the subspace generated by B has dimension at most m+1. Furthermore we prove that in this case, if
, then B is an m-dimensional subspace or a cone with an (m−2)-dimensional vertex over a non-trivial line-blocking set of cardinality q+r2(q)+1 in a plane skew to the vertex. For q=p3h, p7 and q not a square we show this assertion for |B|qm+qm−1+…+q+1+q2/3·(qm−1+…+1). 相似文献
9.
John B. Polhill 《Designs, Codes and Cryptography》2002,25(3):299-309
There have been several recent constructions of partial difference sets (PDSs) using the Galois rings
for p a prime and t any positive integer. This paper presents constructions of partial difference sets in
where p is any prime, and r and t are any positive integers. For the case where
2$$
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many of the partial difference sets are constructed in groups with parameters distinct from other known constructions, and the PDSs are nested. Another construction of Paley partial difference sets is given for the case when p is odd. The constructions make use of character theory and of the structure of the Galois ring
, and in particular, the ring
×
. The paper concludes with some open related problems. 相似文献
10.
Difference Sets Corresponding to a Class of Symmetric Designs 总被引:1,自引:0,他引:1
We study difference sets with parameters(v, k, ) = (p
s(r
2m - 1)/(r - 1), p
s-1
r
2m-2
r - 1)r
2m -2, where r = r
s - 1)/(p - 1) and p is a prime. Examples for such difference sets are known from a construction of McFarland which works for m = 1 and all p,s. We will prove a structural theorem on difference sets with the above parameters; it will include the result, that under the self-conjugacy assumption McFarland's construction yields all difference sets in the underlying groups. We also show that no abelian .160; 54; 18/-difference set exists. Finally, we give a new nonexistence prove of (189, 48, 12)-difference sets in Z
3 × Z
9 × Z
7. 相似文献
11.
We consider strongly regular graphs = (V, E) on an even number, say 2n, of vertices which admit an automorphism group G of order n which has two orbits on V. Such graphs will be called strongly regular semi-Cayley graphs. For instance, the Petersen graph, the Hoffman–Singleton graph, and the triangular graphs T(q) with q 5 mod 8 provide examples which cannot be obtained as Cayley graphs. We give a representation of strongly regular semi-Cayley graphs in terms of suitable triples of elements in the group ring Z
G. By applying characters of G, this approach allows us to obtain interesting nonexistence results if G is Abelian, in particular, if G is cyclic. For instance, if G is cyclic and n is odd, then all examples must have parameters of the form 2n = 4s
2 + 4s + 2, k = 2s
2 + s, = s
2 – 1, and = s
2; examples are known only for s = 1, 2, and 4 (together with a noncyclic example for s = 3). We also apply our results to obtain new conditions for the existence of strongly regular Cayley graphs on an even number of vertices when the underlying group H has an Abelian normal subgroup of index 2. In particular, we show the nonexistence of nontrivial strongly regular Cayley graphs over dihedral and generalized quaternion groups, as well as over two series of non-Abelian 2-groups. Up to now these have been the only general nonexistence results for strongly regular Cayley graphs over non-Abelian groups; only the first of these cases was previously known. 相似文献
12.
We prove the nonexistence of a distance-regular graph with intersection array {74,54,15;1,9,60} and of distance-regular graphs with intersection arrays
{4r3+8r2+6r+1,2r(r+1)(2r+1),2r2+2r+1;1,2r(r+1),(2r+1)(2r2+2r+1)}