Existence of directed GDDs with block size five

Bennett

2010

Des. Codes Cryptogr.

Self Cite

Wang and Yin have established the existence of a directed BIBD with block size 7 and index 1 (a DBIBD(v, 7, 1)) for all v ≡ 1, 7 (mod 21) except for v = 22 and possibly for 68 other cases. In this paper, we reduce the number of possible exceptions to 4, namely v = 274, 358, 400, 526. Correspondingly, for all such v, our results establish the existence of a T(2, 7, v)-code or equivalently a perfect 5-deletion-correcting code with words of length 7 over an alphabet of size v, where all the coordinates must be different. In the process, we also reduce the possible exceptions for (v, 7, 2)-BIBDs to 2 cases, v = 274 and 358 (in addition to the non-existent (22, 7, 2)-BIBD).

Section: Preliminaries and Construction Methodsmentioning

confidence: 99%

“…3,4,5,6,10,14,15,18,20,22,26,30,34,38, 46, 60}. A TD(7, g), or equivalently, a (7, 1)-GDD of type g 7 is known to exist for any g / ∈ E, see for instance, [2].…”

Section: Preliminaries and Construction Methodsmentioning

confidence: 99%

“…( (2,12), (1,5), (0, 26), (1, 24), (2, 2), (0, 1), (1, 18)), ( (1,9), (2, 24), (0, 1), a 0 , (0, 7), (2,12), (1,14)), (1,13), (0, 26), (2,9)). …”

Section: Direct Constructionsunclassified

mentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Existence of directed BIBDs with block size 7 and related perfect 5-deletion-correcting codes of length 7

Bennett

2010

Des. Codes Cryptogr.

Self Cite

Super‐simple holey Steiner pentagon systems and related designs

J of Combinatorial Designs

Bennett

2007

A Steiner pentagon system of order v (SPS(v)) is said to be super-simple if its underlying (v, 5, 2)-BIBD is super-simple; that is, any two blocks of the BIBD intersect in at most two points. It is well known that the existence of a holey Steiner pentagon system (HSPS) of type T implies the existence of a (5, 2)-GDD of type T. We shall call an HSPS of type T super-simple if its underlying (5, 2)-GDD of type T is super-simple; that is, any two blocks of the GDD intersect in at most two points. The existence of HSPSs of uniform type h n has previously been investigated by the authors and others. In this article, we focus our attention on the existence of super-simple HSPSs of uniform type h n .

New results on GDDs, covering, packing and directable designs with block size 5

J of Combinatorial Designs

Assaf

Bluskov

et al. 2010

Self Cite

This article looks at (5, k) GDDs and (v, 5, k) pair packing and pair covering designs. For packing designs, we solve the (4t, 5, 3) class with two possible exceptions, solve 16 open cases with k odd, and improve the maximum number of blocks in some (v, 5, k) packings when v small (here, the Schönheim bound is not always attainable). When k = 1, we construct v = 432 and improve the spectrum for v ≡ 14, 18 (mod 20). We also extend one of Hanani's conditions under which the Schönheim bound cannot be achieved (this extension affects (20t+9, 5, 1), (20t+17, 5, 1) and (20t+13, 5, 3)) packings. For covering designs we find the covering numbers C(280, 5, 1), C(44, 5, 17) and C(44, 5, k) with k ≡ 13 (mod 20). We also know that the covering number, C(v, 5, 2), exceeds the Schönheim bound by 1 for v = 9, 13 and 15. For GDDs of type g n , we have one new design of type 30 9 when k = 1, and three new designs for k = 2, namely, types g 15 with g ∈{13, 17, 19}. If k is even and a (5, k) GDD of type g u is known, then we also have a directable (5, k) GDD of type g u . q NEW RESULTS ON GDDS, COVERING, PACKING AND DIRECTABLE DESIGNS 339 2. AUXILIARY DESIGNS Some of the terminology we will use is quite standard in design theory; see [24]. For clarification of our notation (specifically how we indicate the standard parameters), we refer to pairwise balanced designs (PBDs), (including BIBDs), as (v, K , ) designs, where K is a list of block sizes that possibly occur. A group divisible design is referred to as a (K , ) GDD of group type g t 1 1 , . . . , g t n n if there are t i groups of size g i and transversal designs of order n as TD (k, n), dropping the subscript when = 1; note that a TD (k, n) is a (k, ) GDD of group type n k . The prefix "R" will denote a resolvable design.Theorem 2.1 (Hanani [36]). Necessary conditions for the existence of a (v, k, ) BIBD are that (v −1) ≡ 0 (mod k −1), v(v −1) ≡ 0 (mod k(k −1)) and v ≥ k. These conditions are sufficient when k = 5 with the definite exception of (v, k, ) = (15, 5, 2). Theorem 2.2 (Abel et al. [5]). A TD(5, n) exists for all n ≥ 4 with the definite exception of n = 6 and the possible exception of n = 10. A TD(6, n) exists for all n ≥ 5 with the definite exception of n = 6 and the possible exception of n = 10, 14, 18 and 22.The next theorem is a combination of work by several authors, see for instance, [34,35,58]. One new design, a (5, 1)-GDD of type 30 9 will be given later in Example 9.1. [34,35], Yin et al. Theorem 2.3 (Ge and Ling[58]). The necessary conditions for the existence of a (5, 1) GDD of type g u are sufficient except for g u ∈{2 5 , 2 11 , 3 5 , 6 5 }, and possibly except for a. g u = 3 45 , 3 65 ; b. g ≡ 2, 6, 14, 18 (mod 20) and:(i) g = 2 and u ∈{15, 35, 71, 75, 95, 111, 115, 195, 215}; (ii) g = 6 and u ∈{15, 35, 75, 95}; (iii) g = 18 and u ∈{11, 15, 71, 111, 115}; (iv) g ∈{14, 22, 26, 34, 38, 46, 58, 62} and u ∈{11, 15, 71, 75, 111, 115}; (v) g ∈{42, 54} and u = 15; (vi) g = 2 with gcd(30, ) = 1 and 33 ≤ ≤ 2443, and u = 15; c. g ≡ 10 (mod 20) and: (i) g...