Group divisible designs, GBRDSDS and generalized weighing matrices

We give new constructions for regular group divisible designs, pairwise balanced designs, generalized Bhaskar Rao supplementary difference sets and generalized weighing matrices. In particular if p is a prime power and q divides p - 1 we show the following exist; (i) GDD (2(p2+p+1), 2(p2+p+1), rp2,2p2, λ1 = p2λ, λ2 = (p2-p)r, m=p2+p+1,n=2), r_+1,2; (ii) GDD(q(p+1), q(p+1), p(q-1), p(q-1),λ1=(q-1)(q-2), λ2=(p-1)(q-1)2/q,m=q,n=p+1); (iii) PBD(21,10;K),K={3,6,7} and PDB(78,38;K), K={6,9,45}; (iv) GW(vk,k2;EA(k)) whenever a (v,k,λ)-difference set exists and k is a prime power; (v) PBIBD(vk2,vk2,k2,k2;λ1=0,λ2=λ,λ3=k) whenever a (v,k,λ)-difference set exists and k is a prime power; (vi) we give a GW(21;9;Z3)

Constructions of balanced ternary designs based on generalized Bhaskar Rao designs

New series of balanced ternary designs and partially balanced ternary designs are obtained. Some of the designs in the series are non-isomorphic solutions for design parameters which were previously known or whose solution was obtained by trial and error, rather than by a systematic method

On rr-extendability of the hypercube Q\sb n

summary:A graph having a perfect matching is called $r$-extendable if every matching of size $r$ can be extended to a perfect matching. It is proved that in the hypercube $Q_n$, a matching $S$ with $|S|\leq n$ can be extended to a perfect matching if and only if it does not saturate the neighbourhood of any unsaturated vertex. In particular, $Q_n$ is $r$-extendable for every $r$ with $1\leq r\leq n-1.

Group divisible designs of four groups and block size five with configuration (1; 1; 1; 2)

We present constructions and results about GDDs with four groups and block size five in which each block has Configuration $(1, 1, 1, 2)$, that is, each block has exactly one point from three of the four groups and two points from the fourth group. We provide the necessary conditions of the existence of a GDD$(n, 4, 5; \lambda_1, \lambda_2)$ with Configuration $(1, 1, 1, 2)$, and show that the necessary conditions are sufficient for a GDD$(n, 4, 5; \lambda_1,$ $\lambda_2)$ with Configuration $(1, 1, 1, 2)$ if $n \not \equiv 0 ($mod $6)$, respectively. We also show that a GDD$(n, 4, 5; 2n, 6(n - 1))$ with Configuration $(1, 1, 1, 2)$ exists, and provide constructions for a GDD$(n = 2t, 4, 5; n, 3(n - 1))$ with Configuration $(1, 1, 1, 2)$ where $n \not= 12$, and a GDD$(n = 6t, 4, 5; 4t, 2(6t - 1))$ with Configuration $(1, 1, 1, 2)$ where $n \not= 6$ and $18$, respectively

Encryption methods based on combinatorial designs

We explore the use of some combinatorial designs for possible use as secret codes. We are motivated to use designs as (1) combinatorial designs are often hard to find, (2) the algorithms for encryption ond decryption are of reasonable length, (3) combinatorial designs have very large numbers of designs in each equivalence class lending themselves readily to selection using a secret key

Coloured designs, new group divisible designs and pairwise balanced designs

Many new families of group divisible designs, balanced incomplete block designs and pairwise balanced designs can be obtained by using constructions based on coloured designs (CD). This paper gives one such construction in each case together with an existence theorem for coloured designs

Generalised Bhaskar Rao designs of block size 3 over the group Z4

We show that the necessary conditions (i) 2tv(v-l) ≡ O(mod 3) (ii) v ≥ 3 (iii) t ≡ 1,5 (mod 6) =\u3e v ≠ 3 are sufficient for the existence of a GBRD(v,3,4t;Z4) except possibly when (v,t) = (27,1) or (39,1)

Group Divisible Designs, GBRSDs and Generalized Weighing Matrices

We give new constructions for regular group divisible designs, pairwise balanced designs, generalized Bhaskar Rao supplementary difference sets and generalized weighing matrices. In particular if p is a prime power and q divides p \Gamma 1 we show the following exist: (i) GDD(2(p 2 +p+ 1), 2(p 2 +p+ 1), rp 2 , 2p 2 , 1 = p 2 , 2 = (p 2 \Gamma p)r, m = p 2 +p+ 1, n = 2), r = 1; 2; (ii) GDD(q(p+ 1), q(p + 1), p(q \Gamma 1), p(q \Gamma 1), 1 = (q \Gamma 1)(q \Gamma 2), 2 = (p \Gamma 1)(q \Gamma 1) 2 =q, m = q, n = p + 1); (iii) PBD(21; 10; K), K = f3; 6; 7g and PBD(78; 38; K), K = f6; 9; 45g; (iv) GW (vk; k 2 ; EA(k)) whenever a (v; k; )-difference set exists and k is a prime power; (v) PBIBD(vk 2 ; vk 2 ; k 2 ; k 2 ; 1 = 0, 2 = , 3 = k) whenever a (v; k; )-difference set exists and k is a prime power; (vi) we give a GW (21; 9; Z 3 ). The GDD obtained are not found in W.H. Clatworthy, Tables of Two-Associate-Class, Partially Balanced Designs, NBS, US Department of Commerce, 1971.

A note on GDD(1, n, n, 4; λ_1, λ_2)

The present note is motivated by two papers on group divisible designs (GDDs) with the same block size three but different number of groups: three and four where one group is of size $1$ and the others are of the same size $n$. Here we present some interesting constructions of GDDs with block size 4 and three groups: one of size $1$ and other two of the same size $n$. We also obtain necessary conditions for the existence of such GDDs and prove that they are sufficient in several cases. For example, we show that the necessary conditions are sufficient for the existence of a GDD$(1,n,n,4;\lambda_1,\lambda_2)$ for $n\equiv0,1,4,5,8,9\pmod{12}$ when $\lambda_1\ge \lambda_2$