We study 1-perfect codes in Doob graphsD(m,n). We show that such codes that are linear over GR(4) exist if and only if n = (4γ+δ−1)/3 andm = (4γ+2δ−4γ+δ)/6 for some integers γ ≥ 0 and δ > 0. We also prove necessary conditions on (m,n) for 1-perfect codes that are linear over Z4 (we call such codes additive) to exist in D(m,n) graphs; for some of these parameters, we show the existence of codes....

We study codes with parameters of the ternary Hamming $(n=(3^m-1)/2,3^{n-m},3)$ code, i.e., $1$-perfect codes. The rank code is defined to be dimension its affine span. characterize $n-m+1$, count their number, and prove that all such can obtained from each other by a sequence two-coordinate switchings. enumerate length $13$ concatenation lengths $9$ $4$; we find there are $93241327$ equivalenc...

We consider the well-studied problem of finding a perfect matching in d-regular bipartite graphs with 2n vertices and m = nd edges. While the best-known algorithm for general bipartite graphs (due to Hopcroft and Karp) takes O(m √ n) time, in regular bipartite graphs, a perfect matching is known to be computable in O(m) time. Very recently, the O(m) bound was improved to O(min{m, n 2.5 lnn d })...

We consider the size of circuits that perfectly hash an arbitrary subset S⊂{0, 1} of cardinality 2 into {0, 1}. We observe that, in general, the size of such circuits is exponential in 2k−m, and provide a matching upper bound.

We give various reformulations of the Strong Perfect Graph Conjecture, based on a study of forced coloring procedures, uniquely colorable subgraphs and ! ? 1-cliques in minimal imperfect graphs.

The amount of at retrieval time is one of the when looking for such functioiis. evidence about the effectivensss reduce the memory requirements perfect hash function proposed

We obtain a new upper bound for odd multiperfect numbers. If N is an odd perfect number with k distinct prime divisors and P is its largest prime divisor, we find as a corollary that 1012P 2N < 24 k . Using this new bound, and extensive computations, we derive the inequality k ≥ 10. Introduction One of the oldest unsolved problems in mathematics is whether there exists an odd perfect number N ....

A lterbank-based method of time-scale modi cation is analyzed for elemental signals including clicks, sines, and AM-FM sines. It is shown that with the use of some basic properties of linear systems, as well as FM-to-AM lter transduction, \perfect reconstruction" time-scaling lterbanks can be constructed for these elemental signal classes under certain conditions on the lterbank. Conditions for...