Related papers: How Much String to String a Cardioid?
We introduce stamp chains. A stamp chain is a finite integer sequence that is both an addition chain and an additive 2-basis, i.e., a solution to the postage stamp problem. We provide a simple method for converting known postage stamp…
In this paper, we show infinite series of $3$-designs in the extended quadratic residue codes over $\mathbb {F}_{r^2}$ for a prime $r$.
An integer array y = y[1..n] is said to be feasible if and only if y[1] = n and, for every i \in 2..n, i \le i+y[i] \le n+1. A string is said to be indeterminate if and only if at least one of its elements is a subset of cardinality greater…
Consider a string of $n$ positions, i.e. a discrete string of length $n$. Units of length $k$ are placed at random on this string in such a way that they do not overlap, and as often as possible, i.e. until all spacings between neighboring…
The theory of sequences, supported by many SMT solvers, can model program data types including bounded arrays and lists. Sequences are parameterized by the element data type and provide operations such as accessing elements, concatenation,…
The cornerstone of any algorithm computing all repetitions in a string of length n in O(n) time is the fact that the number of runs (or maximal repetitions) is O(n). We give a simple proof of this result. As a consequence of our approach,…
Repetitive Scenario Design (RSD) is a randomized approach to robust design based on iterating two phases: a standard scenario design phase that uses $N$ scenarios (design samples), followed by randomized feasibility phase that uses $N_o$…
In the present paper, complete designs of graphs are considered. The notion of (regular) sampling is introduced and analyzed in detail, showing that the trivial necessary condition for its existence is actually sufficient. Some examples are…
Classical complexity theory measures the cost of computing a function, but many computational tasks require committing to one valid output among several. We introduce determination depth -- the minimum number of sequential layers of…
Much research in stringology focuses on structures that can, in a way, ``grasp'' repeats (substrings that occur multiple times) as, for example, the so-called runs, a.k.a. maximal repetitions, compactly describe all tandem repeats. In this…
A weakly constrained code is a collection of finite-length strings over a finite alphabet in which certain substrings or patterns occur according to some prescribed frequencies. Buzaglo and Siegel (ITW 2017) gave a construction of weakly…
We try to embed a t-design in a finite commutative group in such a way that the sum of the k points of a block is zero. We can compute the number of blocks of the boolean 2-design having all the non zero vectors of $(Z_2)^n$ as the set of…
Music is a repetition of patterns and rhythms. It can be composed by repeating a certain number of bars in a structured way. In this paper, the objective is to generate a loop of 8 bars that can be used as a building block of music. Even…
Circuit codes are constructed from induced cycles in the graph of the $n$ dimensional hypercube. They are both theoretically and practically important, as circuit codes can be used as error correcting codes. When constructing circuit codes,…
The aim of this paper is to present a recursive construction of simple t-designs for arbitrary t. The construction is of purely combinatorial nature and it requires finding solutions for the indices of the ingredient designs that satisfy a…
In this paper we provide a new locally consistent decomposition of strings. Each string $x$ is decomposed into blocks that can be described by grammars of size $\widetilde{O}(k)$ (using some amount of randomness). If we take two strings $x$…
In order to design and implement tracers, one must decide what exactly to trace and how to produce this trace. On the one hand, trace designs are too often guided by implementation concerns and are not as useful as they should be. On the…
Cyclic codes are an interesting type of linear codes and have applications in communication and storage systems due to their efficient encoding and decoding algorithms. They have been studied for decades and a lot of progress has been made.…
This article introduces recursive relations allowing the calculation of the number of partitions with constraints on the minimum and/or on the maximum fragment size.
Digital System Research has pioneered the mathematics and design for a new class of computing machine using residue numbers. Unlike prior art, the new breakthrough provides methods and apparatus for general purpose computation using several…