Narrow Results

We use means of formal language theory to estimate the Hausdorff measure of sets of a certain shape in Cantor space. These sets are closely related to infinite iterated function systems in fractal geometry.

Our results are used to provide a series of simple examples for the non-coincidence of limit sets and attractors for infinite iterated function systems.

The problem of negative design of DNA languages is addressed, that is, properties and construction methods of large sets of words that prevent undesired bonds when used in DNA computations. We recall a few existing formalizations of the problem and then define the property of sim-bond-freedom, where sim is a similarity relation between words. We show that this property is decidable for context-free languages and polynomial-time decidable for regular languages. The maximality of this property also turns out to be decidable for regular languages and polynomial-time decidable for an important case of the Hamming similarity. Then we consider various construction methods for Hamming bond-free languages, including the recently introduced method of templates, and obtain a complete structural characterization of all maximal Hamming bond-free languages. This result is applicable to the θ-k-code property introduced by Jonoska and Mahalingam.

The branch of coding theory that is based on formal languages has produced several methods for defining code properties, including word relations, dependence systems, implicational conditions, trajectories, and language inequations. Of those, the latter three can be viewed as formal methods in the sense that a certain formal expression can be used to denote a code property. Here we present a formal method which is based on transducers. Each transducer of a certain type defines/describes a desired code property. The method provides simple and uniform decision procedures for the basic questions of property satisfaction and maximality for regular languages. Our work includes statements about the hardness of deciding some of the problems involved. It turns out that maximality can be hard to decide even for "classical" code properties of finite languages. We also present an initial implementation of a LAnguage SERver capable of deciding the satisfaction problem for a given transducer code property and regular language.

A two-dimensional code of pictures is defined as a set X ⊆ Σ** such that any picture over Σ is tilable in at most one way with pictures in X. It is proved that in general it is undecidable whether a finite set of picture is a code. The subclass of prefix codes is introduced and it is proved that it is decidable whether a finite set of pictures is a prefix code. Further a polynomial time decoding algorithm for finite prefix codes is given. Maximality and completeness of finite prefix codes are studied.

This paper deals with insertability and mainly extractablity of codes. A code C is called insertable (or extractable) if the free submonoid C^* generated by C satisfies if z, $xy\in C^{*}$ implies $x\mathrm{zy}\in C^{*}$ (or z, $x\mathrm{zy}\in C^{*}$ implies $xy\in C^{*}$). We show that a finite insertable code is a full uniform code. On the other hand there are many finite extractable codes which are not full uniform codes. We cannot still characterize the structures of infinite extractable codes. Here we give some results on the class of infix extractable codes. First, we consider a necessary and sufficient condition whether a given infix code C is extractable or not by using the syntactic graph of the code. Secondly, we investigate the extractability for the families of other related bifix codes. We newly define the bifix codes, called e(m)-codes and $\overline{e}$-codes, and refer to the extractability of them.

Compressed sensing is a sparse sampling theory. Compared with the Nyquist-Shannon sampling theory, in compressed sensing one could reconstruct a sparse signal from a few linear and non-adaptive measurements. How to construct a good sensing matrix which captures the full information of a sparse signal is an important problem in compressed sensing. In this paper, we present a new deterministic construction using a linear or nonlinear code, which is a generalization of DeVore’s construction and Li et al.’s construction. By choosing some appropriate linear codes or nonlinear codes, we will construct some good binary sensing matrices which are superior to DeVore’s ones and Li et al.’s ones.

We study how the generative power of networks of evolutionary processors decreases if special codes and ideals are used as input and output filters of the nodes instead of arbitrary regular languages. We show that all recursively enumerable languages can be generated if the filters are right ideals or left ideals. With respect to codes, we obtain a hierarchy of evolutionary language families which is almost identical to that of the families of codes.

The $k$-prefix-free, $k$-suffix-free and $k$-infix-free languages generalize the prefix-free, suffix-free and infix-free languages by allowing marginal errors. For example, a string $x$ in a $k$-prefix-free language $L$ can be a prefix of up to $k$ different strings in $L$. We also define finitely prefix-free languages in which a string $x$ can be a prefix of finitely many strings. We present efficient algorithms that determine whether or not a given regular language is $k$-prefix-free, $k$-suffix-free or $k$-infix-free, and analyze the time complexity of the algorithms. We establish undecidability results for deciding these properties for (linear) context-free languages.

In a paper concerning the relationship between coding theory and factorization theory Restivo, Salemi, and Sportelli made a conjecture that if subsets possess certain properties then they cannot form a factorization of a finite cyclic group. In this note it is shown that in fact such factorizations do exist.

In a recent paper, Carrell and Goulden found a combinatorial identity of the Bernstein operators that they then used to prove Bernstein's theorem. We show that this identity is a straightforward consequence of the classical result. We also show how a similar approach using the codes of partitions can be generalized from Schur functions to also include Schur Q-functions and derive the combinatorial formulation for both cases. We then apply them by examining the Littlewood–Richardson and Pieri rules.

The action of the Bernstein operators on Schur functions was given in terms of codes by Carrell and Goulden (2011) and extended to the analog in Schur Q-functions in our previous work. We define a new combinatorial model of extended codes and show that both of these results follow from a natural combinatorial relation induced on codes. The new algebraic structure provides a natural setting for Schur functions indexed by compositions.

In this paper, we study codes where the alphabet is a finite Frobenius bimodule over a finite ring. We discuss the extension property for various weight functions. Employing an entirely character-theoretic approach and a duality theory for partitions on Frobenius bimodules, we derive alternative proofs for the facts that the Hamming weight and the homogeneous weight satisfy the extension property. We also use the same techniques to derive the extension property for other weights, such as the Rosenbloom–Tsfasman weight.

There is a special local ring $E$ of order $4,$ without identity for the multiplication, defined by $E=〈a,b|2a=2b=0,a^2=a,b^2=b,ab=a,ba=b〉.$ We study the algebraic structure of linear codes over that non-commutative local ring, in particular their residue and torsion codes. We introduce the notion of quasi self-dual codes over $E,$ and Type IV codes, that is quasi self-dual codes whose all codewords have even Hamming weight. We study the weight enumerators of these codes by means of invariant theory, and classify them in short lengths.

In a previous paper, the second and third named authors introduced the concept of the complete cycle index and discussed a relation with the complete weight enumerator in coding theory. In this paper, we introduce the concept of the complete joint cycle index and the average complete joint cycle index, and discuss a relation with the complete joint weight enumerator and the average complete joint weight enumerator, respectively, in coding theory. Moreover, the notion of the average intersection numbers is given, and we discuss a relation with the average intersection numbers in coding theory.

Permutation decoding method developed by MacWilliams and described in [Permutation decoding of systematic codes, Bell Syst. Tech. J.43 (1964) 485–505] is a decoding technique that uses a subset of the automorphism group of the code called a PD-set. The complexity of the permutation decoding algorithm depends on the size of the PD-set and finding a minimal PD-set for an error correcting code is a hard problem. In this paper we examine binary codes from the complete-multipartite graph $\mathrm{\Gamma }=K_{n_1,\dots ,n_m}$ and find PD-sets for all values of $m$ and $n_i$. Further we show that these PD-sets are minimal when $m$ is odd and $n_1=n_2=\cdots =n_m$.

This chapter explores the concepts of ethics, morals and social responsibility from organisational and societal perspectives covering both marketing that is focused on profit and marketing focused on bringing about social benefit. It discusses the meanings of social responsibility from different paradigmatic viewpoints and highlights the advantages and limitations of particular approaches. The chapter also considers some aspects of legal and regulatory frameworks and the potential for the development of codes of conduct for socially responsible for-profit marketing and social marketing. The discussion is positioned in a global context and is grounded by intercultural considerations and the diversity of ethical perspectives and norms across cultures.

Algebraic geometrical concepts are playing an increasing role in quantum applications such as coding, cryptography, tomography and computing. We point out here the prominent role played by Galois fields viewed as cyclotomic extensions of the integers modulo a prime characteristic p. They can be used to generate efficient cyclic encoding, for transmitting secrete quantum keys, for quantum state recovery and for error correction in quantum computing. Finite projective planes and their generalization are the geometric counterpart to cyclotomic concepts, their coordinatization involves Galois fields, and they have been used repetitively for enciphering and coding. Finally, the characters over Galois fields are fundamental for generating complete sets of mutually unbiased bases, a generic concept of quantum information processing and quantum entanglement. Gauss sums over Galois fields ensure minimum uncertainty under such protocols. Some Galois rings which are cyclotomic extensions of the integers modulo 4 are also becoming fashionable for their role in time encoding and mutual unbiasedness.

Let ℓ > 0 be a square free integer and the ring of integers of the imaginary quadratic field . Codes C over K determine lattices Λ_ℓ(C) over rings . The theta functions θ_Λℓ(C) of such lattices are known to determine the symmetrized weight enumerator swe(C) for small primes p = 2, 3; see [1, 10].

In this paper we explore such constructions for any p. If p ∤ ℓ then the ring is isomorphic to 𝔽_p² or 𝔽_p × 𝔽_p. Given a code C over we define new theta functions on the corresponding lattices. We prove that the theta series θ_Λℓ(C) can be written in terms of the complete weight enumerator of C and that θ_Λℓ(C) is the same for almost all ℓ. Furthermore, for large enough ℓ, there is a unique complete weight enumerator polynomial which corresponds to θ_Λℓ(C).

The binary self-dual [2ⁿ, 2^n-1, n]₂ codes from the adjacency matrices of the n-cubes Q_n, where n ≥ 6 and is even, are examined and 2- and 3-PD-sets of size n2ⁿ are found.

Scientific discoveries, especially over the last six decades, have left no doubt that ‘information’ plays a central role in biology. Specialists have thus sought to study the information in biological systems using the same definitions of information as have been traditionally used in engineering, computer science, mathematics and in other disciplines. Unfortunately, all of these traditional definitions lack aspects that even non-specialists recognize as being essential attributes of information — qualities such as meaning and purpose. To remedy that deficiency, we define another type of information — Universal Information — that more accurately embodies the full measure of information. We then examine the DNA/RNA protein synthesizing system with this definition of Universal Information and conclude that Universal Information is indeed present and that it is essential for all biological life. Furthermore, other types of information, such as Mental Imaging Information, also play a key role in life. It thus seems inevitable that the biological sciences (and science in general) must consider other-than-the-traditional definitions of information if we are to answer some of the fundamental questions about life.