A binary extended 1-perfect code C folds over its kernel via the Steiner quadruple systems associated with its codewords. The resulting folding, proposed as a graph invariant for C, distinguishes among the 361 nonlinear codes C of kernel dimension κ obtained via Solov’eva-Phelps doubling construction, where 9 ≥ κ ≥ 5. Each of the 361 resulting graphs has most of its nonloop edges expressible in...

A linear (qd, q, t)-perfect hash family of size s in a vector space V of order qd over a field F of order q consists of a set φ1, . . . , φs of linear functionals from V to F with the following property: for all t subsets X ⊆ V there exists i ∈ {1, . . . , s} such that φi is injective when restricted to F . A linear (qd, q, t)-perfect hash family of minimal size d(t− 1) is said to be optimal. I...

Super-simple designs are useful in constructing codes and designs such as superimposed codes and perfect hash families. Recently, Gronau et al determined the existence of super-simple (v, 5, 2)-BIBDs with possible exceptions of v ∈ {75, 95, 115, 135, 195, 215, 231, 285, 365, 385, 515}. In this article, we investigate the existence of a super-simple (v, 5, 4)-BIBD and show that such a design exi...

We present two new perfect hashing schemes that can be used for efficient bitboard move generation for sliding pieces in chess-like board games without the need to use rotated bitboards. Moreover, we show that simple variations of these schemes give minimal perfect hashing schemes. The new method is applicable provided N , the number of k-bit spaced positions that may be set to 1, is not more t...

The construction of perfect dominating sets in all binary 2-cubes (r > 3) with the same number of edges along each coordinate direction is performed, thus generalizing the known construction of this for r = 3. However, an adaptation of that construction aiming to perfect dominating sets in the 14-ternary cube with edges along all coordinate directions still fails. 1 Perfect Dominating Sets in B...

We present a N-dimensional generalization of the two-dimensional block-circulant perfect array construction by [Blake, 2013]. As in [Blake, 2013], the families of N-dimensional arrays possess pairwise good zero correlation zone (ZCZ) cross-correlation. Both constructions use a perfect autocorre-lation sequence with the array orthogonality property (AOP). This paper presents a generalization of ...

It is proven that for any numbers n = 2 m − 1, m ≥ 4 and r, such that n − log(n + 1) ≤ r ≤ n excluding n = r = 63, n = 127, r ∈ {126, 127} and n = r = 2047 there exists a propelinear perfect binary code of length n and rank r.

We introduce the concept of the ghost dimension gh. dim. R of a ring R. This is the longest nontrivial chain of maps in the derived category emanating from a perfect complex such that each map is zero on homology. We show that gh. dim. R ≤ w. dim. R, with equality if R is coherent or w. dim. R = 1.

An (s; n; q; t)-perfect hash family is a set of functions 1 ; 2 ; : : :; s from a set V of cardinality n to a set F of cardinality q with the property that every t-subset of V is injectively mapped into F by at least one of the functions i. The paper shows that the maximum value n s;t (q) that n can take for xed s and t has a leading term that is linear in q if and only if t > s. Moreover, for ...