Related papers: On Universal Codes for Integers: Wallace Tree, Eli…
This paper investigates universal polar coding schemes. In particular, a notion of ordering (called convolutional path) is introduced between probability distributions to determine when a polar compression (or communication) scheme designed…
We describe a combinatorial approach for investigating properties of rational numbers. The overall approach rests on structural bijections between rational numbers and familiar combinatorial objects, namely rooted trees. We emphasize that…
In 1975 Chaitin introduced his \Omega number as a concrete example of random real. The real \Omega is defined based on the set of all halting inputs for an optimal prefix-free machine U, which is a universal decoding algorithm used to…
Good integers introduced in 1997 form an interesting family of integers that has been continuously studied due to their rich number theoretical properties and wide applications. In this paper, we have focused on classes of $2^\beta$-good…
Routing tables in ad hoc and wireless routing protocols can be represented using rooted trees. The constant need for communication and storage of these trees in routing protocols demands an efficient rooted tree coding algorithm. This…
Recently, the weight distributions of the duals of the cyclic codes with two zeros have been obtained for several cases. In this paper we provide a slightly different approach toward the general problem and use it to solve one more special…
In this work, we study the Codeword Stabilized Quantum Codes (CWS codes) a generalization of the stabilizers quantum codes using a new approach, the algebraic structure of modules, a generalization of linear spaces. We show then a new…
The notion of transducer integer sequences is considered through a series of examples. By definition, transducer integer sequences are integer sequences produced, under a suitable interpretation, by finite automata encoding tree morphisms…
We provide a complete description of the Wadge hierarchy for deterministically recognisable sets of infinite trees. In particular we give an elementary procedure to decide if one deterministic tree language is continuously reducible to…
We present a novel technique for encoding and decoding constant weight binary codes that uses a geometric interpretation of the codebook. Our technique is based on embedding the codebook in a Euclidean space of dimension equal to the weight…
Bounds on the entropy of patterns of sequences generated by independently identically distributed (i.i.d.) sources are derived. A pattern is a sequence of indices that contains all consecutive integer indices in increasing order of first…
In this paper we give a fast algorithm to generate all partitions of a positive integer $n$. Integer partitions may be encoded as either ascending or descending compositions for the purposes of systematic generation. It is known that the…
In this paper, a family of five-weight reducible cyclic codes is presented. Furthermore, the weight distribution of these cyclic codes is determined, which follows from the determination of value distributions of certain exponential sums.
A pattern of a sequence is a sequence of integer indices with each index describing the order of first occurrence of the respective symbol in the original sequence. In a recent paper, tight general bounds on the block entropy of patterns of…
Consider a finite positive integer. If it is even, divide it by 2, and if it is odd, multiply it by 3 and add 1. This will give you a new integer. Following the procedure for the new integer, you will receive another integer. Repeat the…
Natural language text corpora are often available as sets of syntactically parsed trees. A wide range of expressive tree queries are possible over such parsed trees that open a new avenue in searching over natural language text. They not…
Subspace codes and particularly constant dimension codes have attracted much attention in recent years due to their applications in random network coding. As a particular subclass of subspace codes, cyclic subspace codes have additional…
A fruitful way of obtaining meaningful, possibly concrete, algorithmically random numbers is to consider a potential behaviour of a Turing machine and its probability with respect to a measure (or semi-measure) on the input space of binary…
Motivated by a concept studied in [1], we consider a property of matrices over finite fields that generalizes triangular totally nonsingular matrices to block matrices. We show that (1) matrices with this property suffice to construct good…
Distributions on integers are ubiquitous in probabilistic modeling but remain challenging for many of today's probabilistic programming languages (PPLs). The core challenge comes from discrete structure: many of today's PPL inference…