A construction of D-optimal designs for N≡2 mod 4

D-optimal design of order 6 is used to construct D-optimal designs of order 42 and 66.     

Simple 3-designs and PSL(2, q) with q ≡ 1 (mod 4)

In this paper, we consider the action of (2, q) on the finite projective line for q ≡ 1 (mod 4) and construct several infinite families of simple 3-designs which admit PSL(2, q) as an automorphism group. Some of the designs are also minimal. We also indicate a general outline to obtain some other algebraic constructions of simple 3-designs.      

Some new 3-designs from PSL(2, q) with q ≡ 1 (mod4)

We determine the sizes of orbits from the action of subgroups of PSL(2,q) on projective line X = GF(q) ∪ {∞} with q a prime power and congruent to 1 modulo 4.As an example of its application,we construct some new families of simple 3-designs admitting PSL(2,q) as automorphism group.     

On a property of symmetric designs of order n≡2 (mod 4)

Summary U. Ott, during his visit in Rome (spring 1985), by using the theory of even unimodular lattices, proved that a (v,k,) symmetric design of order n2 (mod 4) satisfies the congruence v ±1 (mod 8). He asked me the question whether this is a consequence of the Bruck-Ryser-Chowla's theorem or not. In this paper we prove that the answer to this question is affirmative. As a consequence of this, we have that the conjecture according to which the Bruck-Ryser-Chowla's theorem and the identity k²–v=n imply the existence of a (v,k,) symmetric design is still open.     

实二次域Q(√p)(p≡3(mod4))类数的上界

设p是适合p≡3(mod4)的奇素数,h,分别是实二次域Q(√p)的类数和基本单位.本文运用初等方法证明了:εh<(p+a+2)a+2/4(a+2)!,其中a=[(√p+1)/2].     

Generalized Steiner Triple Systems with Group Size g ≡0, 3 (mod 6)

Generalized Steirier triple systems, GS(2,3,n,g), are equivalent to maximum constant weight codes over an alphabet of size g 1 with distance 3 and weight 3 in which each codeword has length n. The necessary conditions for the existence of a GS(2,3,n,g) are (n-1)g≡0 (mod 2), n(n-1)g2≡0 (mod 6), and n≥g 2. These necessary conditions are shown to be sufficient by several authors for 2≤g≤11. In this paper, three new results are obtained. First, it is shown that for any given g, g≡0 (mod 6) and g≥12, if there exists a GS(2.3.n.g) for all n, g 2≤n≤7g 13. then the necessary conditions are also sufficient. Next, it is also shown that for any given g, g≡3 (mod 6) and g≥15, if there exists a GS(2,3,n,g) for all n, n≡1 (mod 2) and g 2≤n≤7g 6, then the necessary conditions are also sufficient. Finally, as an application, it is proved that the necessary conditions for the existence of a GS(2,3,n,g) are also sufficient for g=12,15.     

实二次域Q(P(1/2))(p≡3(mod 4))类数的上界

设p是适合p≡3（Pod4）的奇素数,h,ε分别是实二次域Q的类数和基本单位．本文运用初等方法证明了：εh＜（p a 2)a 2/4(a 2)!,其中     

实二次域Q(P(1/2))(p≡3(mod 4))类数的上界

设p是适合p≡3（Pod4）的奇素数,h,ε分别是实二次域Q的类数和基本单位．本文运用初等方法证明了：εh＜（p+a+2)a+2/4(a+2)!,其中     

A new extension of the (H.2) supercongruence of Van Hamme for primes $$pequiv 3pmod {4}$$ p ≡ 3 ( mod 4 )

The Ramanujan Journal - Using Andrews’ multiseries generalization of Watson’s $$_8phi _7$$ transformation, we give a new extension of the (H.2) supercongruence of Van Hamme for primes...     

On the Congruence σ(n) ≡ 1(mod n), II

Let k ≥ 2 be an integer, and let σ(n) denote the sum of the positive divisors of an integer n. We call n a quasi-multiperfect number if σ(n) = kn + 1. In this paper, we give some necessary properties of quasi-multiperfect numbers with four different prime divisors.     

Pkn(k≡2(mod3))的D(2)-点可区别全染色

图的D(β)-点可区别全染色就是指图G的一个正常全染色且使得距离不大于β的任意两点有不同的色集合.讨论了幂图P_n~k当k≡2(mod3)时的D(2)-的点可区别全染色,并且根据P_n~2与C_n~2图的结构关系获得C_n~2的邻点可区别的全染色数.     

Determinantenabschätzung für binäre Matrizen mitn≡3 mod 4

Ohne Zusammenfassung     

Pkn(k≡2(mod 3))的邻点可区别的强全染色

对简单图G(V,E),V(Gk)=V(G),E(Gk)＝E(G)U{uv|d(u,v)=k},称Gk为G的k次方图,其中d(u,v)表示u,v在G中的距离.设f为用k色时G的正常全染色法,对 uv∈E(G),满足C(u)≠C(v),其中C(u)＝{f(u)}U{f(v)|uv∈E(G)}U{f(uv)|uv∈E(G)},则称f为G的k邻点可区别的强全染色法,简记作k-ASVDTC,且称Xast(G)＝min{k|k-ASVDTC ofG}为G的邻点可区别的强全色数.本文得到了k≡2(mod 3)时的Xast(Pkn),其中Pn为n阶路.     

P_n~k(k≡2(mod3))的邻点可区别的强全染色

对简单图 G(V,E) ,V(Gk) =V(G) ,E(Gk ) =E(G)∪ { uv|d(u,v) =k} ,称 Gk为 G的 k次方图 ,其中d (u,v)表示 u,v在 G中的距离 .设 f为用 k色时 G的正常全染色法 ,对 uv∈ E(G) ,满足 C(u)≠ C(v) ,其中C(u) ={ f(u) }∪ { f(v) |uv∈ E(G) }∪ { f(uv) |uv∈ E(G) } ,则称 f 为 G的 k邻点可区别的强全染色法 ,简记作 k- ASVDTC,且称 χast(G) =min{ k|k- ASVDTC of G}为 G的邻点可区别的强全色数 .本文得到了 k≡2 (mod3)时的 χast(Pkn) ,其中 Pn 为 n阶路 .     

Generalized Steiner Systems GS(2, 4, v, 2) with v a Prime Power ≡ 7 (mod 12)

Generalized Steiner systems GS(2, 4, v, g) were first introduced by Etzion and were used to construct optimal constant weight codes over an alphabet of size g + 1 with minimum Hamming distance 5, in which each codeword has length v and weight 4. Etzion conjectured that the necessary conditions v 1 (mod 3) and v ; 7 are also sufficient for the existence of a GS(2,4,v,2). Except for the example of a GS(2,4,10,2) and some recursive constructions given by Etzion, nothing else is known about this conjecture. In this paper, Weil's theorem on character sum estimates is used to show that the conjecture is true for any prime power v 7 (mod 12) except v = 7, for which there does not exist a GS(2,4,7,2).     

论二次欧氏域■d~(1/2)(d≡3(mod4)中代数整数β有原根的条件

我们知道二次域K=(d~(1/d))(为有理数域为欧氏域,当且仅当d 为(参看文献[7]): -11;-7;-3;-2;-1;2;3;5;6;7;11; 13;17;19;21;29;33;37;41;57;73。以O_k表示K=(d~(1/d))的代数整数环。文献[1]、[2]和[3]分别讨论了当d=-1,±2和3时,O_k内整数有原根的条件。本文将讨论,对任意的二次欧氏域K=(d~(1/d))(d≡3(mod4))中代数整数β有原根的条件,此时,d=-1;3;7;11或19。