Binary Words Of Length Research Articles

A short Note on Graphs Functionality

The practicality of a graph G is wherever the most is seized all elicited sub graphs H of Functionality is outlined by analogy with degeneracy, that it generalizes: if we tend to replace with within the on top of definition, we tend to get the degeneracy of G. Taking the most over elicited sub graphs ensures that practicality ne'er will increase once taking elicited sub graphs. Equally to several different graph parameters, the notion of graph practicality becomes valuable once it’s worth is tiny, i.e., is finite by a continuing freelance of the dimensions of the graph. Above all, graphs of tiny practicality admit compact illustration, as was shown in [3]. That paper doesn't formally outline the notion of graph practicality; however the results proved there imply that graphs of finite practicality may be depicted by binary words of length

New bounds on antipowers in words

Fici et al. defined a word to be a k-power if it is the concatenation of k consecutive identical blocks, and an r-antipower if it is the concatenation of r pairwise distinct blocks of the same size. They defined N(k,r) as the smallest ℓ such that every binary word of length ℓ contains either a k-power or an r-antipower. In this note we obtain some new upper and lower bounds on N(k,r). We also consider avoiding 3-antipowers and 4-antipowers over larger alphabets, and obtain a lower bound for N(k,5) in the binary case.

On semi-perfect de Bruijn words

We show an application of Lempel's recursive construction of De Bruijn words to the generation of binary words having many factor-rich prefixes. A binary word is said to be factor-rich iff it has the largest number of distinct factors among binary words with the same length. A linear de Bruijn word of rank n is a shortest word containing (as a factor) exactly once each binary word of length n. It is factor-rich and its length equals Δn=2n+n−1. We construct for each n a binary linear de Bruijn word of rank n which is semi-perfect in the following sense: each of its prefixes of length m>Δn−1 is factor-rich. The number Δn−1 is the best possible (for n>2 there is no linear binary de Bruijn word with factor-rich prefix of length m=Δn−1). We show an efficient algorithm constructing compact description of binary semi-perfect de Bruijn words.

Enumerating lambda terms by weighted length of their De Bruijn representation

John Tromp introduced the so-called ‘binary lambda calculus’ as a way to encode lambda terms in terms of 0–1-strings using the de Bruijn representation along with a weighting scheme. Later, Grygiel and Lescanne conjectured that the number of binary lambda terms with m free indices and of size n (encoded as binary words of length n and according to Tromp’s weights) is o(n−3∕2τ−n) for τ≈1.963448…. We generalize the proposed notion of size and show that for several classes of lambda terms, including binary lambda terms with m free indices, the number of terms of size n is Θn−3∕2ρ−n with some class dependent constant ρ, which in particular disproves the above mentioned conjecture.The methodology used is setting up the generating functions for the classes of lambda terms. These are infinitely nested radicals which are investigated then by a singularity analysis.We show further how some properties of random lambda terms can be analyzed and present a way to sample lambda terms uniformly at random in a very efficient way. This allows to generate terms of size more than one million within a reasonable time, which is significantly better than the samplers presented in the literature so far.

On distance Gray codes

A Gray code of size n is a cyclic sequence of all binary words of length n such that two consecutive words differ exactly in one position. We say that the Gray code is a distance code if the Hamming distance between words located at distance k from each other is equal to d. The distance property generalizes the familiar concepts of a locally balanced Gray code. We prove that there are no distance Gray codes with d = 1 for k > 1. Some examples of constructing distance Gray codes are given. For one infinite series of parameters, it is proved that there are no distance Gray codes.

Fibonacci [formula omitted]-cubes as Cartesian products

The Fibonacci (p,r)-cube Γn(p,r) is the subgraph of Qn induced on binary words of length n in which there are at most r consecutive ones and there are at least p zeros between two substrings of ones. These cubes simultaneously generalize several interconnection networks, notably hypercubes, Fibonacci cubes, and postal networks. In this note it is proved that Γn(p,r) is a non-trivial Cartesian product if and only if p=1 and r=n≥2, or p=r=2 and n≥2, or n=p=3 and r=2. This rounds a result from Ou et al. (2011) asserting that Γn(2,2) are non-trivial Cartesian products.

A regularity lemma and twins in words

For a word S, let f(S) be the largest integer m such that there are two disjoint identical (scattered) subwords of length m. Let f(n,Σ)=min{f(S):S is of length n, over alphabet Σ}. Here, it is shown that2f(n,{0,1})=n−o(n) using the regularity lemma for words. In other words, any binary word of length n can be split into two identical subwords (referred to as twins) and, perhaps, a remaining subword of length o(n). A similar result is proven for k identical subwords of a word over an alphabet with at most k letters.

On the number of frames in binary words

A frame is a square u u , where u is an unbordered word. Let F ( n ) denote the maximum number of distinct frames in a binary word of length n . We count this number for small values of n and show that F ( n ) is at most ⌊ n / 2 ⌋ + 8 for all n and greater than 7 n / 30 − ϵ for any positive ϵ and infinitely many n . We also show that Fibonacci words, which are known to contain plenty of distinct squares, have only a few frames. Moreover, by modifying the Thue–Morse word, we prove that the minimum number of occurrences of frames in a word of length n is ⌈ n / 2 ⌉ − 2 .

A Bound on Equitable Partitions of the Hamming Space

We denote Qn the set of binary words of length n. A partition {C1,C2,?,Cr} of Qn with quotient matrix B = (bij)r×r is equitable if for all i and j, any word in d has exactly bij neighbors in Cj. The equitable partitions of Qn can be obtained from completely regular codes. We derive a bound on equitable partitions of Qn that does not depend on the size of the partition.

Spectral and symbolic analysis of heart rate data during the tilt test

Spectral analysis of heart rate sequences is commonly used to investigate neuroauthonomic control of heart rate by means of two indexes, the low and the high frequency power. For tilt test data of normal subjects we compare the spectral indexes with new indexes defined within the framework of symbolic analysis. We define two classes of binary words of length 4: the first class is related to "acceleration" of heart rate and the second class to "stationary behavior." The new indexes measure the change in frequency of the two classes before and after the tilt. Data analysis of 13 normal subjects shows that the behavior of the new indexes is in agreement with that of spectral ones.

On fragments of words

We find a precise value of the function F N(m, n, k), which is the number of binary words of length N and weight m that contain an arbitrary word of length n and weight k as a fragment. As a consequence, we obtain a known result on the number of binary words of length N that contain a fixed word of length n as a fragment.

Polynomial versus exponential growth in repetition-free binary words

It is known that the number of overlap-free binary words of length n grows polynomially, while the number of cubefree binary words grows exponentially. We show that the dividing line between polynomial and exponential growth is 7 3 . More precisely, there are only polynomially many binary words of length n that avoid 7 3 -powers, but there are exponentially many binary words of length n that avoid 7 3 + -powers. This answers an open question of Kobayashi from 1986.

Short Dominating Paths and Cycles in the Binary Hypercube

A sequence of binary words of length n is called a cube dominating path, if the Ham-ming distance between two consecutive words is always one, and every binary word of length n is within Hamming distance one from at least one of these words. If also the first and last words are Hamming distance one apart, the sequence is called a cube dominating cycle. Bounds on the cardinality of such sequences are given, and it is shown that asymptotically the shortest cube dominating path and cycle consist of 2^n (1 + o(1))/n words.

Construction of Certain Cyclic Distance-Preserving CodesHaving Linear-Algebraic Characteristics

An ordered list of binary words of length n is called a distance-preserving \langle m,n\rangle-code, if the list distance between two words is equal to their Hamming distance, for distances up to m. A technique for constructing cyclic \langle m,n\rangle-codes is presented, based on the standard Gray code and on some simple tools from linear algebra.

On the normality of multiple covering codes

A binary code C of length n is called a μ-fold r-covering if every binary word of length n is within Hamming distance r of at least μ codewords of C. The normality and the amalgamated direct sum (ADS) construction of 1-fold coverings have been extensively studied. In this paper we generalize the concepts of subnormality and normality to μ-fold coverings and discuss how the ADS construction can be applied to them. In particular, we show that for r = 1, 2 all binary linear μ-fold r-coverings of length at least 2 r + 1 and μ-fold normal.