Steven T. Dougherty

• Ph.D.
• University of Scranton

Additive codes were initially introduced by Delsarte in 1973 within the context of association schemes and recently they have become of interest due to their application in constructing quantum error-correcting codes. We give foundational results for additive codes where the elements are from a finite field, and define the orthogonality relation...

In this study, the utilization of exponentiation in finite fields is investigated for the purpose of generating pseudo-random sequences which have a crucial role in cryptographic applications. More precisely, a novel method for generating pseudo-random sequences is proposed to construct an initial S-Box which is a key component in various encryptio...

This paper presents an image encryption algorithm by using time signature-dependent S-Boxes, which are based on Latin squares, the Playfair system of cryptography, and functions that are inspired by the behavior of a Japanese ladder. The encryption algorithm includes four stages: the construction of the S-Box, the generation of keys, image diffusio...

We relate quasi-self-dual codes over a non-unital ring to additive self-dual codes over the additive group of the ring. We show that the Hamming weight enumerator of a quasi-self-dual code over this non-unital ring must belong to the same ring of invariants that self-dual codes over the Frobenius rings with unities belong. Additionally, the Hamming...

In this work, we present a new construction method for reversible codes. We employ composite matrices derived from group rings and show how to construct these matrices so that they are also reversible. Also in this work, we give an algorithm for calculating conflict free DNA codes that satisfy the Hamming distance, the reverse, the reverse-compleme...

In this work, we present a matrix construction for reversible codes derived from skew dihedral group rings. By employing this matrix construction, the ring \begin{document}$ \mathcal{F}_{j, k} $\end{document} and its associated Gray maps, we show how one can construct reversible codes of length \begin{document}$ n2^{j+k} $\end{document} over the fi...

In this paper, we give a new method for constructing LCD codes. We employ group rings and a well known map that sends group ring elements to a subring of the $n \times n$ matrices to obtain LCD codes. Our construction method guarantees that our LCD codes are also group codes, namely, the codes are ideals in a group ring. We show that with a certain...

We present a generator matrix of the form \begin{document}$ [ \sigma(v_1) \ | \ \sigma(v_2)] $\end{document}, where \begin{document}$ v_1 \in RG $\end{document} and \begin{document}$ v_2\in RH $\end{document}, for finite groups \begin{document}$ G $\end{document} and \begin{document}$ H $\end{document} of order \begin{document}$ n $\end{document} f...

In this paper, we introduce double Quadratic Residue Codes (QRC) of length \(n=p+q\) for prime numbers p and q in the ambient space \({{\mathbb {F}}} _{2}^{p}\times {{\mathbb {F}}}_{2}^{q}.\) We give the structure of separable and non-separable double QRC over this alphabet and we show that interesting double QR codes in this space exist only in th...

Additive codes have become an increasingly important topic in algebraic coding theory due to their applications in quantum error-correction and quantum computing. Linear Complementary Dual (LCD) codes play an important role for improving the security of information against certain attacks. Motivated by these facts, we define additive complementary...

In this paper, we show that one can construct a \begin{document}$ G $\end{document}-code from group rings that is reversible. Specifically, we show that given a group with a subgroup of order half the order of the ambient group with an element that is its own inverse outside the subgroup, we can give an ordering of the group elements for which \beg...

In this work, we study a new family of rings, B_{j,k}, whose base field is the finite field F_{p^r}. We study the structure of this family of rings and show that each member of the family is a commutative Frobenius ring. We define a Gray map for the new family of rings, study G-codes, self-dual G-codes, and reversible G-codes over this family. In p...

In this paper, we give a new method for constructing LCD codes. We employ group rings and a well known map that sends group ring elements to a subring of the $n \times n$ matrices to obtain LCD codes. Our construction method guarantees that our LCD codes are also group codes, namely, the codes are ideals in a group ring. We show that with a certain...

A subset of a vector space $\mathbb {F}_{q}^{n}$ is additive if it is a linear space over the field $\mathbb {F}_{p}$ , where $q=p^{e}$ , $p$ prime, and $e>1$ . Bounds on the rank and dimension of the kernel of additive generalised Hadamard (additive GH) codes are established. For specific ranks and dimensions of the kernel within these b...

In this work, we define composite matrices which are derived from group rings. We extend the idea of G-codes to composite G-codes. We show that these codes are ideals in a group ring, where the ring is a finite commutative Frobenius ring and G is an arbitrary finite group. We prove that the dual of a composite G-code is also a composite G-code. We...

We construct an infinite family of commutative rings Rq,Δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${R_{q,\varDelta }}$$\end{document} and we study codes over thes...

We define additive G-codes over finite fields. We prove that if C is an additive G-code over Fq with duality M then its dual with respect to this duality CM is an additive G-code. We prove that if M and M′ are two dualities, then CM and CM′ are equivalent codes. Finally, we study the existence of self-dual codes for a variety of dualities and relat...

In this work, we study codes generated by elements that come from group matrix rings. We present a matrix construction which we use to generate codes in two different ambient spaces: the matrix ring M_k(R) and the ring R, where R is the commutative Frobenius ring. We show that codes over the ring M_k(R) are one sided ideals in the group matrix ring...

In this work, we extend an established isomorphism between group rings and a subring of the n by n matrices. This extension allows us to construct more complex matrices over the ring R. We present many interesting examples of complex matrices constructed directly from our extension. We also show that some of the matrices used in the literature befo...

In this work, we study codes generated by elements that come from group matrix rings. We present a matrix construction which we use to generate codes in two different ambient spaces: the matrix ring $M_k(R)$ and the ring $R,$ where $R$ is the commutative Frobenius ring. We show that codes over the ring $M_k(R)$ are one sided ideals in the group mat...

We define a self-dual code over a finite abelian group in terms of an arbitrary duality on the ambient space. We determine when additive self-dual codes exist over abelian groups for any duality and describe various constructions for these codes. We prove that there must exist self-dual codes under any duality for codes over a finite abelian group...

This chapter introduces the fundamentals of cryptology. It describes the basic combinatorial principles involved in substitution ciphers and the German Enigma machine. It then develops the main public-key encryption systems including RSA, El Gamal, and the McEliece cryptographic system based on error-correcting codes.

We describe a composite construction from group rings where the groups have orders 16 and 8. This construction is then applied to find the extremal binary self-dual codes with parameters [32, 16, 8] or [32, 16, 6]. We also extend this composite construction by expanding the search field which enables us to find more extremal binary self-dual codes...

Self-dual codes, which are codes that are equal to their orthogonal, are a widely studied family of codes. Various techniques involving circulant matrices and matrices from group rings have been used to construct such codes. Moreover, families of rings have been used, together with a Gray map, to construct binary self-dual codes. In this paper, we...

In this chapter, we shall use one of the most important structures in abstract algebra as a tool to study finite incidence structures. The algebraic structure is a group. It is generally the first structure one encounters in studying abstract algebra. We shall begin with a very elementary study of finite groups, and then we shall study the groups a...

This chapter gives a brief description of discrete probability. It uses the combinatorial counting properties developed earlier in the text to compute various probabilities.

This chapter describes a series of combinatorial objects including Hadamard matrices, Latin hypercubes, association schemes, and partially ordered sets. The algebraic and combinatorial properties of these objects are discussed.

Chapter 5 gives foundational results on graph theory including a study of simple and directed graphs. It investigates the coloring of graphs and the connection between directed graphs and relations.

This chapter introduces a version of the well-known Tic-Tac-Toe game which can be played on designs and finite geometries. This game helps develop students’ geometric intuition. The theory of combinatorial games is applied to determine when the first player has a winning strategy and when the second player can force a draw.

This chapter gives a basic introduction of linear algebra and uses this setting to describe higher dimensional affine and projective geometries. It includes proofs of the Bruck–Ryser theorem and Desargues’ theorem. It further describes Baer subplanes, arcs, and ovals. It concludes with a description of certain non-Desarguesian planes.

This chapter gives fundamental results on finite affine and projective planes. It provides detailed proofs on various counting results concerning these planes such as the number of points, lines, points on a line, and lines through a point. It describes the canonical relation between affine planes and mutually orthogonal Latin squares.

This chapter contains foundational combinatorial results. It covers basic abstract counting techniques and gives a detailed description of permutations of finite sets, relating them to Japanese ladders. It concludes with a description of generating functions, and how they are used to count objects.

Algebraic coding theory arose in the last half of the twentieth century to answer applications in the field of electronic communication. Specifically, the idea is to transmit information over a channel such that, when it is received, whatever errors that were made can be corrected with a high degree of certainty. Since its initial study of this app...

This chapter introduces the topic of finite combinatorial designs. The defining parameters of the designs are determined and their restrictions are proved. Special attention is given to Steiner triple systems, nets, and biplanes.

This chapter gives a complete description of the necessary algebraic techniques for combinatorics. It gives a detailed explanation of modular arithmetic including Fermat’s Little Theorem and Euler’s generalization. It gives foundational results on finite fields to prepare the reader for their use in finite geometry. It concludes with a description...

This chapter describes mutually orthogonal Latin squares by beginning with their origins in the 36 officer problem. It describes the major open problems concerning Latin squares. Further results are given describing the structure of Latin squares.

In this paper, we are interested in finding an algebraic structure of conjucyclic codes of length n over the finite field F4. We show that conjucyclic codes of length n over F4 are related to binary cyclic codes of length 2n and show that there is a canonical bijective correspondence between the two sets. We illustrate how the factorization of the...

Linear complementary dual codes were defined by Massey in 1992, and were used to give an optimum linear coding solution for the two user binary adder channel. In this paper, we define the analog of LCD codes over fields in the ambient space with mixed binary and quaternary alphabets. These codes are additive, in the sense that they are additive sub...

A subset of a vector space $\mathbb{F}_q^n$ is $K$-additive if it is a linear space over the subfield $K\subseteq \mathbb{F}_q$. Let $q=p^e$, $p$ prime, and $e>1$. Bounds on the rank and dimension of the kernel of generalised Hadamard (GH) codes which are $\mathbb{F}_p$-additive are established. For specific ranks and dimensions of the kernel withi...

We introduce skew cyclic and skew constacyclic codes over the local Frobenius non-chain rings of order 16 by defining non-trivial automorphisms on these rings. We study the Gray images of these codes, obtaining a number of binary and quaternary codes with good parameters as images of skew cyclic codes over these rings.

In this paper, we introduce a new bordered construction for self-dual codes using group rings. We consider constructions over the binary field, the family of rings Rk and the ring F4+uF4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepac...

In this work, we study construction methods for self-dual and formally self-dual codes from group rings, arising from the cyclic group, the dihedral group, the dicyclic group and the semi-dihedral group. Using these constructions over the rings F_2+uF_2 and F_4+uF_4, we obtain 9 new extremal binary self-dual codes of length 68 and 25 even formally...

This undergraduate textbook is suitable for introductory classes in combinatorics and related topics. The book covers a wide range of both pure and applied combinatorics, beginning with the very basics of enumeration and then going on to Latin squares, graphs and designs. The latter topic is closely related to finite geometry, which is developed in...

In this work, we define $G$-codes over the infinite ring $R_\infty$ as ideals in the group ring $R_\infty G$. We show that the dual of a $G$-code is again a $G$-code in this setting. We study the projections and lifts of $G$-codes over the finite chain rings and over the formal power series rings respectively. We extend known results of constructin...

We describe eight composite constructions from group rings where the orders of the groups are 4 and 8, which are then applied to find self-dual codes of length 16 over F_4. These codes have binary images with parameters [32,16,8] or [32,16,6]. These are lifted to codes over F_4+uF_4, to obtain codes with Gray images of extremal self-dual binary cod...

Codes over commutative Frobenius rings are studied with a focus on local Frobenius rings of order 16 for illustration. The main purpose of this work is to present a method for constructing a generating character for any commutative Frobenius ring. Given such a character, the MacWilliams identities for the complete and symmetrized weight enumerators...

For any positive odd integer n, a precise representation for cyclic codes over \({\mathbb {Z}}_4\) of length 2n is given in terms of the Chinese Remainder Theorem. Using this representation, an efficient encoder for each of these codes is described. Then the dual codes are determined precisely and this is used to study codes which are self-dual. In...

Linear complementary dual codes were defined by Massey in 1992, and were used to give an optimum linear coding solution for the two user binary adder channel. In this paper, we define the analog of LCD codes over fields in the ambient space with mixed binary and quaternary alphabets. These codes are additive, in the sense that they are additive sub...

We introduce a bordered construction over group rings for self-dual codes. We apply the constructions over the binary field and the ring F_2+uF_2, using groups of orders 9, 15, 21, 25, 27, 33 and 35 to find extremal binary self-dual codes of lengths 20, 32, 40, 44, 52, 56, 64, 68, 88 and best known binary self-dual codes of length 72. In particular...

Conjucyclic codes were first introduced by Calderbank, Rains, Shor and Sloane in [1] because of their applications in quantum error-correction. In this paper, we study linear and additive conjucyclic codes over the finite field \begin{document}$ {\mathbb{F}}_{4} $\end{document} and produce a duality for which the orthogonal, with respect to that du...

We give constructions of self-dual and formally self-dual codes from group rings where the ring is a finite commutative Frobenius ring. We improve the existing construction given in \cite{Hurley1} by showing that one of the conditions given in the theorem is unnecessary and moreover it restricts the number of self-dual codes obtained by the constru...

This presentation is related to the paper: S. T. Dougherty, J.-L. Kim, B. Ö., L. Sok and P. Solé, “The combinatorics of LCD codes: Linear programming bound and orthogonal matrices”, Int. J. Inf. and Coding Theory, vol. 4, no. 2/3, 116-128, 2017. The slides clarify some short proofs in Sections 3 and 4. An explicit construction is given for the pr...

In this work, we study constacyclic codes of odd length over finite commutative local Frobenius non-chain rings of order 16. We give the structure of λi-constacyclic codes over each local Frobenius non-chain ring via cyclic codes over these rings where λi acts as a weight preserving unit in the corresponding ring. We also obtain binary optimal code...

A Z2Z4-additive code C subset of Z_2^alpha x Z_4^beta is called cyclic if the set of coordinates can be partitioned into two subsets, the set of Z_2 and the set of Z_4 coordinates, such that any cyclic shift of the coordinates of both subsets leaves the code invariant. Let Phi(C) be the binary Gray image of C. We study the rank and the dimension of...

A Z2Z4-additive code C subset of Z_2^alpha x Z_4^beta is called cyclic if the set of coordinates can be partitioned into two subsets, the set of Z_2 and the set of Z_4 coordinates, such that any cyclic shift of the coordinates of both subsets leaves the code invariant. Let Phi(C) be the binary Gray image of C. We study the rank and the dimension of...

A Z2Z4-additive code C Z α 2× Z 4 β is called cyclic if the set of coordinates can be partitioned into two subsets, the set of Z2 and the set of Z4 coordinates, such that any cyclic shift of the coordinates of both subsets leaves the code invariant. We study the binary images of Z2Z4-additive cyclic codes. We determine all Z2Z4-additive cyclic code...

A Z2Z4-additive code C is called cyclic if the set of coordinates can be partitioned into two subsets, the set of Z_2 and the set of Z_4 coordinates, such that any cyclic shift of the coordinates of both subsets leaves the code invariant. We study the binary images of Z2Z4-additive cyclic codes. We determine all Z2Z4-additive cyclic codes with odd...

In this chapter, we describe families of rings including the rings of order 4, their generalizations, X-rings, and the ring \(R_{q,\varDelta }\). We describe the kernel and rank of binary codes that are images of quaternary codes via the Gray map. Then a generalized Singleton bound is proven for codes over Frobenius rings.

In this Chapter, we prove the MacWilliams relations for codes over finite Frobenius commutative rings. These relations are one of the foundational results of algebraic coding theory. We describe them first for codes over groups and extend this to codes over Frobenius rings. Finally, we give a practical guide for producing MacWilliams relations for...

In this chapter, we study polycyclic, negacyclic, constacyclic, quasicyclic and skew cyclic codes which are all generalizations of the important family of cyclic codes. We describe their algebraic setting and show how to use this setting to classify these families of codes.

We give the necessary definitions and foundational results from commutative ring theory for the study of codes over rings. We give the definition of Frobenius rings and characterize them in terms of characters. We prove the generalized Chinese Remainder Theorem and describe what constitutes a minimal generating set for a code over a finite Frobeniu...

In this chapter, we describe self-dual codes over Frobenius rings. We give constructions of self-dual codes over any Frobenius ring. We describe connections to unimodular lattices, binary self-dual codes and to designs. We also describe linear complementary dual codes and make a new definition of a broad generalization encompassing both self-dual a...

We make a connection between the braid group and signed permutations. Using this link, we describe a commutative diagram which contains the fundamental sequence for the braid group.

We study cyclic codes over commutative local Frobenius rings of order 16 and give their binary images under a Gray map which is a generalization of the Gray maps on the rings of order 4. We prove that the binary images of cyclic codes are quasi-cyclic codes of index 4 and give examples of cyclic codes of various lengths constructed from these techn...

We study codes over the finite sub Hopf algebras of the Steenrod algebra. We define three dualities for codes over these rings, namely the Eulidean duality, the Hermitian duality and a duality based on the underlying additive group structure. We study self-dual codes, namely codes equal to their orthogonal, with respect to all three dualities.

Linear complementary dual (LCD) codes are binary linear codes that meet their dual trivially. We construct LCD codes using orthogonal matrices, self-dual codes, combinatorial designs and Gray map from codes over the family of rings Rk . We give a linear programming bound on the largest size of an LCD code of given length and minimum distance. We ma...

This book provides a self-contained introduction to algebraic coding theory over finite Frobenius rings. It is the first to offer a comprehensive account on the subject. Coding theory has its origins in the engineering problem of effective electronic communication where the alphabet is generally the binary field. Since its inception, it has grown a...

We study the rank and kernel of (Formula presented.) cyclic codes of odd length n and give bounds on the size of the kernel and the rank. Given that a cyclic code of odd length is of the form (Formula presented.), where (Formula presented.), we show that (Formula presented.) and (Formula presented.) where (Formula presented.) is the preimage of the...

We shall describe several families of X-rings and construct self-dual and formally self-dual codes over these rings. We then use a Gray map to construct binary formally self-dual codes from these codes. In several cases, we produce binary formally self-dual codes with larger minimum distances than known self-dual codes. We also produce non-linear c...

Polycyclic codes are ideals in quotients of polynomial rings by a principal ideal. Special cases are cyclic and constacyclic codes. A MacWilliams relation between such a code and its annihilator ideal is derived. An infinite family of binary self-dual codes that are also formally self-dual in the classical sense is exhibited. We show that right pol...

We define a class of finite Frobenius rings of order , describe their generating characters, and study codes over these rings. We define two conjugate weight preserving Gray maps to the binary space and study the images of linear codes under these maps. This structure couches existing Gray maps, which are a foundational idea in codes over rings, in...

We give an algebraic structure for a large family of binary quasi-cyclic codes. We construct a family of commutative rings and a canonical Gray map such that cyclic codes over this family of rings produce quasi-cyclic codes of arbitrary index in the Hamming space via the Gray map. We use the Gray map to produce optimal linear codes that are quasi-c...

We study self-dual codes over non-commutative Frobenius rings. It is shown that a code is equal to its left orthogonal if and only if it is equal to its right orthogonal. Constructions of self-dual codes are given over Frobenius rings that arise from self-dual codes over the center of the ring. These constructions are used to show for which lengths...

We give constructions of self-dual and formally self-dual codes from group rings where the ring is a finite commutative Frobenius ring. We improve the existing construction given in \cite{Hurley1} by showing that one of the conditions given in the theorem is unnecessary and moreover it restricts the number of self-dual codes obtained by the constru...

We study codes over the commutative local Frobenius rings of order 16 with maximal ideals of size 8. We define a weight preserving Gray map and study the images of these codes as binary codes. We study self-dual codes and determine when they exist.

We study ΘS–cyclic codes over the family of rings Ak We characterize ΘS–cyclic codes in terms of their binary images. A family of Hermitian inner-products is defined and we prove that if a code is ΘS–cyclic then its Hermitian dual is also ΘS–cyclic. Finally, we give constructions of ΘS–cyclic codes.

We study $\Theta_S-$cyclic codes over the family of rings $A_k.$ We characterize $\Theta_S-$cyclic codes in terms of their binary images. A family of Hermitian inner-products is defined and we prove that if a code is $\Theta_S-$cyclic then its Hermitian dual is also $\Theta_S-$cyclic. Finally, we give constructions of $\Theta_S-$cyclic codes.

We consider codes over ℤps with the extended Lee weight. We find Singleton bounds with respect to this weight and define MLDS and MLDR codes accordingly. We also consider the kernels of these codes and the notion of independence of vectors in this space. We investigate the linearity and duality of the Gray images of codes over ℤps.

We study one weight \(\mathbb {Z}_2\mathbb {Z}_4\) additive codes. It is shown that the image of an equidistant \(\mathbb {Z}_2\mathbb {Z}_4\) code is a binary equidistant code and that the image of a one weight \(\mathbb {Z}_2\mathbb {Z}_4\) additive code, with nontrivial binary part, is a linear binary one weight code. The structure and possible...

The ranks and kernels of generalized Hadamard matrices are studied. It is proven that any generalized Hadamard matrix $H(q,\lambda)$ over $F_q$, $q>3$, or $q=3$ and $\gcd(3,\lambda)\not =1$, generates a self-orthogonal code. This result puts a natural upper bound on the rank of the generalized Hadamard matrices. Lower and upper bounds are given for...

Linear Complementary Dual codes (LCD) are binary linear codes that meet their dual trivially. We construct LCD codes using orthogonal matrices, self-dual codes, combinatorial designs and Gray map from codes over the family of rings $R_k$. We give a linear programming bound on the largest size of an LCD code of given length and minimum distance. We...

Codes over commutative Frobenius rings are studied with a focus on local Frobenius rings of order 16 for illustration. The main purpose of this work is to present a method for constructing a generating character for any commutative Frobenius ring. Given such a character, the MacWilliams identities for the complete and symmetrized weight enumerators...

It is well known that the subsets of the Hamming scheme with an abelian group structure have been charcterized by Delsarte as what is now known Z2Z4 codes. Z2Z4-additive codes give rise to Z2Z4-linear codes via a Gray map which are propelinear binary codes. In this paper, we define free Z2Z4-additive codes and count their number. We then count the...

It is well known that the subsets of the Hamming scheme with an abelian group structure have been charcterized by Delsarte as what is now known Z2Z4 codes. Z2Z4-additive codes give rise to Z2Z4-linear codes via a Gray map which are propelinear binary codes. In this paper, we define free Z2Z4-additive codes and count their number. We then count the...

It is well known that the subsets of the Hamming scheme with an abelian group structure have been charcterized by Delsarte as what is now known Z2Z4 codes. Z2Z4-additive codes give rise to Z2Z4-linear codes via a Gray map which are propelinear binary codes. In this paper, we define free Z2Z4-additive codes and count their number. We then count the...

Codes over an infinite family of rings which are an extension of the binary field are defined. Two Gray maps to the binary field are attached and are shown to be conjugate. Euclidean and Hermitian self-dual codes are related to binary self-dual and formally self-dual codes, giving a construction of formally self-dual codes from a collection of arbi...

We study odd and even $\mathbb{Z }_2\mathbb{Z }_4$ formally self-dual codes. The images of these codes are binary codes whose weight enumerators are that of a formally self-dual code but may not be linear. Three constructions are given for formally self-dual codes and existence theorems are given for codes of each type defined in the paper.