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Somphong Jitman

Publications and source records attributed to Somphong Jitman.

At least 19 recordsLinked to original sources
arXiv2026

Good Integers: (T,k)-Subclasses and Applications to Galois Duality in Coding Theory

The notion of good integers, namely the divisors of the sequence $(a^s+b^s)_{s\ge 1}$ for nonzero coprime integers $a$ and $b$, together with their subfamilies such as oddly-good and evenly-good integers, has become an important arithmetic tool in the study of Euclidean and Hermitian dualities for abelian and cyclic codes. Building on this perspective, this paper introduces and studies another interesting subclass of good integers arising from the sequence $\bigl(a^{ks+T}+b^{ks+T}\bigr)_{s\ge 1}$ for some integers $0\leq T<k$, whose divisors are called $(T,k)$-{\em good integers with respect to} $(a,b)$. An arithmetic theory of these integers is developed, including a characterization at odd prime powers, a general characterization for odd integers in terms of $2$-adic valuations, and a treatment of even integers. An explicit algorithm is also given for deciding whether a given integer $d$ is $(T,k)$-good with respect to $(a,b)$ and, when it is, for computing an exponent $s$ such that $d\mid \bigl(a^{ks+T}+b^{ks+T}\bigr)$. Applications in coding theory are then obtained from the specialization $(a,b)=(q,1)$, where $q$ is a prime power. In particular, the $q^k$-cyclotomic classes of the cyclic group $\mathbb Z_n$ characterize the Galois self-reciprocal irreducible factors of $x^n-1$ over $\F_{q^k}$, give a description and enumeration of Galois LCD cyclic codes of length $n$ over $\F_{q^k}$, and lead to a characterization of Galois self-dual cyclic codes.

math.NT↗
arXiv2026

Explicit Determinants of Homogeneous Polynomial Evaluation Matrices and Applications

In this work, the determinants of matrices constructed by evaluating homogeneous bivariate polynomials at pairs of vectors are investigated. For a polynomial $p(x,y)=\sum\limits_{i=0}^k \alpha_i x^{k-i}y^i$, an explicit factorization of the determinant of the associated $n\times n$ evaluation matrix $A_{\mathbf{a},\mathbf{b}}(p(x,y))=\bigl[p(a_r,b_s)\bigr]_{r,s=1}^n$ is presented for all $n \ge k+1$ and for all pairs of vectors $\mathbf a=(a_1,\dots,a_n)$ and $\mathbf b=(b_1,\dots,b_n)$ of length $n$. In particular, it is proved that $\det (A_{\mathbf{a},\mathbf{b}}(p(x,y)))=0$ when $n \ge k+2$, while in the borderline case $n=k+1$ a closed formula involving Vandermonde determinants is derived in the vector sets and the coefficients of $p(x,y)$. Several well-known determinants, including those arising from $(x+y)^k$ and classical quotient forms $\frac{a^k-b^k}{a-b}$ and $\frac{a^k+b^k}{a+b}$, emerge as special cases. We also provide a discussion for $n \le k$, connecting the problem to symmetric functions and generalized Vandermonde determinants. Finally, applications such matrices and determinants are provided, including an explicit formula and equivariance law under linear changes of variables for the sum-form \(p(x,y)=f(x+y)\), and a non-vanishing bound over finite fields via Schwartz-Zippel lemma.

math.RA↗
arXiv2025

Good Integers: A Concise Completion of the Non-Coprime Case

For coprime nonzero integers $a$ and $b$, a positive integer $\ell$ is said to be {\em good} with respect to $a$ and $b$ if there exists a positive integer $k$ such that $\ell |(a^{k}+b^{k})$. Since the early 1990s, such classical good integers have been studied intensively for their number theoretic structures and for applications, notably in coding theory. This work completes the study by relaxing the coprimality hypothesis and treating the non-coprime case $\gcd(a,b)\neq1$ in a concise and self-contained way. The results are presented in terms of the classical coprime criterion and $p$-adic valuations of $\ell$. As a consequence, whenever $\ell$ is good, all admissible exponents form a single arithmetic progression with an explicit starting point and period. Some special cases are discussed in the non-coprime setting. A practical decision procedure is developed that decides the goodness of a given integer and explicitly enumerates the full set of admissible exponents. Several illustrative examples are presented.

math.NT↗
arXiv2025

On the Enumeration of Symmetric Tridiagonal Matrices with prescribed Determinant over Commutative Finite Chain Rings

Determinants of structured matrices play a fundamental role in both pure and applied mathematics, with wide-ranging applications in linear algebra, combinatorics, coding theory, and numerical analysis. In this work, the enumeration of symmetric tridiagonal matrices with prescribed determinant over finite fields and over commutative finite chain rings is developed. Using the recurrent formula for determinants, a recursive form of the numbers of singular and nonsingular symmetric tridiagonal matrices is derived, after which a uniform counting framework for any fixed determinant value is obtained. Over finite fields, quadratic-character techniques are employed. In odd dimensions, the enumeration is shown to be independent of the chosen nonzero determinant. Whereas, in even dimensions, it depends only on the quadratic residue/nonresidue class in the fields. For commutative finite chain rings, explicit formulas for nonsingular case are produced by lifting along the ideal chain and analyzing reduction to the residue field, yielding closed expressions in terms of the nilpotency index and the size of the residue fields. A layered enumeration has been developed for the study of symmetric tridiagonal matrices over commutative finite chain rings by stratifying determinants into ideal and punctured layers. Entry-wise reduction to quotient rings expresses each layer through zero-determinant counts on quotients, yielding formulas for prescribed determinants, including quadratic/non-quadratic residue factors, and a decomposition for the singular ones.

math.RA↗
arXiv2025

Good Integers: A Comprehensive Review with Applications

For nonzero coprime integers $a$ and $b$, a positive integer $\ell$ is said to be \emph{good with respect to $a$ and $b$} if there exists a positive integer $k$ such that $\ell$ divides $a^{k} + b^{k}$. The concept of good integers has been the subject of continuous investigation since the 1990s due to their elegant number-theoretic properties and their significant applications in various areas, particularly in coding theory. This paper provides a comprehensive review of good integers, emphasizing both their theoretical foundations and their practical implications. We first revisit the fundamental number-theoretic properties of good integers and present their characterizations in a systematic manner. The exposition is enriched with well-structured algorithms and illustrative diagrams that facilitate their computation and classification. Subsequently, we explore applications of good integers in the study of algebraic coding theory. In particular, their roles in the characterization, construction, and enumeration of self-dual cyclic codes and complementary dual cyclic codes are discussed in detail. Several examples are provided to demonstrate the applicability of the theory. This review not only consolidates existing results but also highlights the unifying role of good integers in bridging number theory and coding theory.

math.NT↗
arXiv2024

Characterizations and Constructions of Linear Intersection Pairs of Cyclic Codes over Finite Fields

Linear intersection pairs of linear codes have become of interest due to their nice algebraic properties and wide applications. In this paper, we focus on linear intersection pairs of cyclic codes over finite fields. Some properties of cyclotomic cosets in cyclic groups are presented as key tools in the study of such linear intersection pairs. Characterization and constructions of two cyclic codes of a fixed intersecting dimension are given in terms of their generator polynomials and cyclotomic cosets. In some cases, constructions of two cyclic codes of a fixed intersecting subcode are presented as well. Based on the theoretical characterization, some numerical examples of linear intersection pairs of cyclic codes with good parameters are illustrated.

cs.IT↗
arXiv2020

Revisiting the Factorization of $x^n+1$ over Finite Fields

The polynomial $x^n+1$ over finite fields has been of interest due to its applications in the study of negacyclic codes over finite fields. In this paper, a rigorous treatment of the factorization of $x^n+1$ over finite fields is given as well as its applications. Explicit and recursive methods for factorizing $x^n+1$ over finite fields are provided together with the enumeration formula. As applications, some families of negacyclic codes are revisited with more clear and simpler forms.

math.NT↗
arXiv2020

Determinants of some Special Matrices over Commutative Finite Chain Rings

Circulant matrices over finite fields and over commutative finite chain rings have been of interest due to their nice algebraic structures and wide applications. In many cases, such matrices over rings have a closed connection with diagonal matrices over their extension rings. In this paper, the determinants of diagonal and circulant matrices over commutative finite chain rings $R$ with residue field $\mathbb{F}_q$ are studied. The number of $n\times n$ diagonal matrices over ${R}$ of determinant $a$ is determined for all elements $a$ in $ {R}$ and for all positive integers $n$. Subsequently, the enumeration of nonsingular $n\times n$ circulant matrices over ${R}$ of determinant $a$ is given for all units $a$ in $ {R}$ and all positive integers $n$ such that $\gcd(n,q)=1$. In some cases, the number of singular $n\times n$ circulant matrices over ${R}$ with a fixed determinant is determined through the link between the rings of circulant matrices and diagonal matrices. As applications, a brief discussion on the determinant of diagonal and circulant matrices over commutative finite principal ideal rings is given. Finally, some open problems and conjectures are posted

math.RA↗
arXiv2019

Hulls of Linear Codes Revisited with Applications

Hulls of linear codes have been of interest and extensively studied due to their rich algebraic structures and wide applications. In this paper, alternative characterizations of hulls of linear codes are given as well as their applications. Properties of hulls of linear codes are given in terms of their Gramians of their generator and parity-check matrices. Moreover, it is show that the Gramian of a generator matrix of every linear code over a finite field of odd characteristic is diagonalizable. Subsequently, it is shown that a linear code over a finite field of odd characteristic is complementary dual if and only if it has an orthogonal basis. Based on this characterization, constructions of good entanglement-assisted quantum error-correcting codes are provided.

cs.IT↗
arXiv2019

SRIM and SCRIM Factors of $x^n+1$ over Finite Fields and Their Applications

Self-Reciprocal Irreducible Monic (SRIM) and Self-Conjugate-Reciprocal Irreducible Monic (SRCIM) factors of $x^n-1$ over finite fields have become of interest due to their rich algebraic structures and wide applications. In this paper, these notions are extended to factors of $x^n+ 1$ over finite fields. Characterization and enumeration of SRIM and SCRIM factors of $x^n+1$ over finite fields are established. Simplification and recessive formulas for the number of such factors are given. Finally, applications in the studied of complementary negacyclic codes are discussed.

cs.IT↗
arXiv2019

Self-Dual Linear Codes over $\mathbb{F}_{q}+u\mathbb{F}_{q}+u^2\mathbb{F}_{q}$ and Their Applications in the Study of Quasi-Abelian Codes

Self-dual codes over finite fields and over some finite rings have been of interest and extensively studied due to their nice algebraic structures and wide applications. Recently, characterization and enumeration of Euclidean self-dual linear codes over the ring~$\mathbb{F}_{q}+u\mathbb{F}_{q}+u^2\mathbb{F}_{q}$ with $u^3=0$ have been established. In this paper, Hermitian self-dual linear codes over $\mathbb{F}_{q}+u\mathbb{F}_{q}+u^2\mathbb{F}_{q}$ are studied for all square prime powers~$q$. Complete characterization and enumeration of such codes are given. Subsequently, algebraic characterization of $H$-quasi-abelian codes in $\mathbb{F}_q[G]$ is studied, where $H\leq G$ are finite abelian groups and $\mathbb{F}_q[H]$ is a principal ideal group algebra. General characterization and enumeration of $H$-quasi-abelian codes and self-dual $H$-quasi-abelian codes in $\mathbb{F}_q[G]$ are given. For the special case where the field characteristic is $3$, an explicit formula for the number of self-dual $A\times \mathbb{Z}_3$-quasi-abelian codes in $\mathbb{F}_{3^m}[A\times \mathbb{Z}_3\times B]$ is determined for all finite abelian groups $A$ and $B$ such that $3\nmid |A|$ as well as their construction. Precisely, such codes can be represented in terms of linear codes and self-dual linear codes over $\mathbb{F}_{3^m}+u\mathbb{F}_{3^m}+u^2\mathbb{F}_{3^m}$.

cs.IT↗
arXiv2019

An efficient method to construct self-dual cyclic codes of length $p^s$ over $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$

Let $p$ be an odd prime number, $\mathbb{F}_{p^m}$ be a finite field of cardinality $p^m$ and $s$ a positive integer. Using some combinatorial identities, we obtain certain properties for Kronecker product of matrices over $\mathbb{F}_p$ with a specific type. On that basis, we give an explicit representation and enumeration for all distinct self-dual cyclic codes of length $p^s$ over the finite chain ring $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$ $(u^2=0)$. Moreover, We provide an efficient method to construct every self-dual cyclic code of length $p^s$ over $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$ precisely.

cs.IT↗
arXiv2019

Some Generalizations of Good Integers and Their Applications in the Study of Self-Dual Negacyclic Codes

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 $2^β$-good integers, $2^β$-oddly-good integers, and $2^β$-evenly-good integers which are generalizations of good integers. Properties of such integers have been given as well as their applications in characterizing and enumerating self-dual negacyclic codes over finite fields. An alternative proof for the characterization of the existence of a self-dual negacyclic code over finite fields has been given in terms of such generalized good integers. A general enumeration formula for the number of self-dual negacyclic codes of length $n$ over finite fields has been established. For some specific lengths, explicit formulas have been provided as well. Some known results on self-dual negacyclic codes over finite fields can be formalized and viewed as special cases of this work.

cs.IT↗
arXiv2019

Linear $\ell$-Intersection Pairs of Codes and Their Applications

In this paper, a linear $\ell$-intersection pair of codes is introduced as a generalization of linear complementary pairs of codes. Two linear codes are said to be a linear $\ell$-intersection pair if their intersection has dimension $\ell$. Characterizations and constructions of such pairs of codes are given in terms of the corresponding generator and parity-check matrices. Linear $\ell$-intersection pairs of MDS codes over $\mathbb{F}_q$ of length up to $q+1$ are given for all possible parameters. As an application, linear $\ell$-intersection pairs of codes are used to construct entanglement-assisted quantum error correcting codes. This provides a large number of new MDS entanglement-assisted quantum error correcting codes.

cs.IT↗
arXiv2019

Hulls of Cyclic Codes over $\mathbb{Z}_4$

The hulls of linear and cyclic codes over finite fields have been of interest and extensively studied due to their wide applications. In this paper, the hulls of cyclic codes of length $n$ over the ring $\mathbb{Z}_4$ have been focused on. Their characterization has been established in terms of the generators viewed as ideals in the quotient ring $\mathbb{Z}_4[x]/\langle x^n-1\rangle$. An algorithm for computing the types of the hulls of cyclic codes of arbitrary odd length over $\mathbb{Z}_4$ has been given. The average $2$-dimension $E(n)$ of the hulls of cyclic codes of odd length $n$ over $\mathbb{Z}_4$ has been established. A general formula for $E(n)$ has been provided together with its upper and lower bounds. It turns out that $E(n)$ grows the same rate as $n$.

cs.IT↗
arXiv2019

Self-Dual and Complementary Dual Abelian Codes over Galois Rings

Self-dual and complementary dual cyclic/abelian codes over finite fields form important classes of linear codes that have been extensively studied due to their rich algebraic structures and wide applications. In this paper, abelian codes over Galois rings are studied in terms of the ideals in the group ring ${\rm GR}(p^r,s)[G]$, where $G$ is a finite abelian group and ${\rm GR}(p^r,s)$ is a Galois ring. Characterizations of self-dual abelian codes have been given together with necessary and sufficient conditions for the existence of a self-dual abelian code in ${\rm GR}(p^r,s)[G]$. A general formula for the number of such self-dual codes is established. In the case where $\gcd(|G|,p)=1$, the number of self-dual abelian codes in ${\rm GR}(p^r,s)[G]$ is completely and explicitly determined. Applying known results on cyclic codes of length $p^a$ over ${\rm GR}(p^2,s)$, an explicit formula for the number of self-dual abelian codes in ${\rm GR}(p^2,s)[G]$ are given, where the Sylow $p$-subgroup of $G$ is cyclic. Subsequently, the characterization and enumeration of complementary dual abelian codes in ${\rm GR}(p^r,s)[G]$ are established. The analogous results for self-dual and complementary dual cyclic codes over Galois rings are therefore obtained as corollaries.

math.RA↗
arXiv2018

An explicit representation and enumeration for self-dual cyclic codes over $\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}$ of length $2^s$

Let $\mathbb{F}_{2^m}$ be a finite field of cardinality $2^m$ and $s$ a positive integer. Using properties for Kronecker product of matrices and calculation for linear equations over $\mathbb{F}_{2^m}$, an efficient method for the construction of all distinct self-dual cyclic codes with length $2^s$ over the finite chain ring $\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}$ $(u^2=0)$ is provided. On that basis, an explicit representation for every self-dual cyclic code of length $2^s$ over $\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}$ and an exact formula to count the number of all these self-dual cyclic codes are given.

cs.IT↗