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Patanee Udomkavanich

Publications and source records attributed to Patanee Udomkavanich.

12 recordsLinked to original sources
arXiv2020

Multidimensional Fibonacci Coding

Fibonacci codes are self-synchronizing variable-length codes that are proven useful for their robustness and compression capability. Asymptotically, these codes provide better compression efficiency as the order of the underlying Fibonacci sequence increases, but at the price of the increased suffix length. We propose a circumvention to this problem by introducing higher-dimensional Fibonacci codes for integer vectors. In the process, we provide extensive theoretical background and generalize the theorem of Zeckendorf to higher order. As thus, our work unify several variations of Zeckendorf's theorem while also providing new grounds for its legitimacy.

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↗
arXiv2018

Self-Conjugate-Reciprocal Irreducible Monic Factors of $x^n-1$ over Finite Fields and Their Applications

Self-reciprocal and self-conjugate-reciprocal polynomials over finite fields have been of interest due to their rich algebraic structures and wide applications. Self-reciprocal irreducible monic factors of $x^n-1$ over finite fields and their applications have been quite well studied. In this paper, self-conjugate-reciprocal irreducible monic (SCRIM) factors of $x^n-1$ over finite fields of square order have been focused on. The characterization of such factors is given together the enumeration formula. In many cases, recursive formulas for the number of SCRIM factors of $x^n-1$ have been given as well. As applications, Hermitian complementary dual codes over finite fields and Hermitian self-dual cyclic codes over finite chain rings of prime characteristic have been discussed.

cs.IT↗
arXiv2017

Characterization and Enumeration of Complementary Dual Abelian Codes

Abelian codes and complementary dual codes form important classes of linear codes that have been extensively studied due to their rich algebraic structures and wide applications. In this paper, a family of abelian codes with complementary dual in a group algebra $\mathbb{F}_{p^ν}[G]$ has been studied under both the Euclidean and Hermitian inner products, where $p$ is a prime, $ν$ is a positive integer, and $G$ is an arbitrary finite abelian group. Based on the discrete Fourier transform decomposition for semi-simple group algebras and properties of ideas in local group algebras, the characterization of such codes have been given. Subsequently, the number of complementary dual abelian codes in $\mathbb{F}_{p^ν}[G]$ has been shown to be independent of the Sylow $p$-subgroup of $G$ and it has been completely determined for every finite abelian group $G$. In some cases, a simplified formula for the enumeration has been provided as well. The known results for cyclic complementary dual codes can be viewed as corollaries.

cs.IT↗
arXiv2017

Determinants of Matrices over Commutative Finite Principal Ideal Rings

In this paper, the determinants of $n\times n$ matrices over commutative finite chain rings and over commutative finite principal ideal rings are studied. The number of $n\times n$ matrices over a commutative finite chain ring ${R}$ of a fixed determinant $a$ is determined for all $a\in {R}$ and positive integers $n$. Using the fact that every commutative finite principal ideal ring is a product of commutative finite chain rings, the number of $n\times n$ matrices of a fixed determinant over a commutative finite principal ideal ring is shown to be multiplicative, and hence, it can be determined. These results generalize the case of matrices over the ring of integers modulo $m$.

math.RA↗
arXiv2016

Self-Dual Abelian Codes in some Non-Principal Ideal Group Algebras

The main focus of this paper is the complete enumeration of self-dual abelian codes in non-principal ideal group algebras $\mathbb{F}_{2^k}[A\times \mathbb{Z}_2\times \mathbb{Z}_{2^s}]$ with respect to both the Euclidean and Hermitian inner products, where $k$ and $s$ are positive integers and $A$ is an abelian group of odd order. Based on the well-know characterization of Euclidean and Hermitian self-dual abelian codes, we show that such enumeration can be obtained in terms of a suitable product of the number of cyclic codes, the number of Euclidean self-dual cyclic codes, and the number of Hermitian self-dual cyclic codes of length $2^s$ over some Galois extensions of the ring $\mathbb{F}_{2^k}+u\mathbb{F}_{2^k}$, where $u^2=0$. Subsequently, general results on the characterization and enumeration of cyclic codes and self-dual codes of length $p^s$ over $\mathbb{F}_{p^k}+u\mathbb{F}_{p^k}$ are given. Combining these results, the complete enumeration of self-dual abelian codes in $\mathbb{F}_{2^k}[A\times \mathbb{Z}_2\times \mathbb{Z}_{2^s}]$ is therefore obtained.

math.RA↗
arXiv2016

One-Generator Quasi-Abelian Codes Revisited

The class of 1-generator quasi-abelian codes over finite fields is revisited. Alternative and explicit characterization and enumeration of such codes are given. An algorithm to find all 1-generator quasi-abelian codes is provided. Two 1-generator quasi-abelian codes whose minimum distances are improved from Grassl's online table are presented.

math.RA↗
arXiv2016

Constacyclic and Quasi-Twisted Hermitian Self-Dual Codes over Finite Fields

Constacyclic and quasi-twisted Hermitian self-dual codes over finite fields are studied. An algorithm for factorizing $x^n-λ$ over $\mathbb{F}_{q^2}$ is given, where $λ$ is a unit in $\mathbb{F}_{q^2}$. Based on this factorization, the dimensions of the Hermitian hulls of $λ$-constacyclic codes of length $n$ over $\mathbb{F}_{q^2}$ are determined. The characterization and enumeration of constacyclic Hermitian self-dual (resp., complementary dual) codes of length $n$ over $\mathbb{F}_{q^2}$ are given through their Hermitian hulls. Subsequently, a new family of MDS constacyclic Hermitian self-dual codes over $\mathbb{F}_{q^2}$ is introduced. As a generalization of constacyclic codes, quasi-twisted Hermitian self-dual codes are studied. Using the factorization of $x^n-λ$ and the Chinese Remainder Theorem, quasi-twisted codes can be viewed as a product of linear codes of shorter length some over extension fields of $\mathbb{F}_{q^2}$. Necessary and sufficient conditions for quasi-twisted codes to be Hermitian self-dual are given. The enumeration of such self-dual codes is determined as well.

math.RA↗
arXiv2012

A Classification of Regressive Transformation Semigroups on Chains

For each subchain $X'$ of a chain $X$, let $T_{RE}(X, X')$ denote the semigroup under composition of all full regressive transformations, $α:X\rightarrow X'$ satisfying $xα\leq x$ for all $x\in X$. Necessary and sufficient conditions for $T_{RE}(X,X')$ and $T_{RE}(Y,Y')$ to be isomorphic are given. This isomorphism theorem is applied to classify the semigroup of regressive transformations $T_{RE}(X,X')$ where $X$ are familiar subchains of $\R$, the chain of real numbers.

math.RA↗
arXiv2010

Skew Constacyclic Codes over Finite Chain Rings

Skew polynomial rings over finite fields ([7] and [10]) and over Galois rings ([8]) have been used to study codes. In this paper, we extend this concept to finite chain rings. Properties of skew constacyclic codes generated by monic right divisors of $x^n-λ$, where $λ$ is a unit element, are exhibited. When $λ^2=1$, the generators of Euclidean and Hermitian dual codes of such codes are determined together with necessary and sufficient conditions for them to be Euclidean and Hermitian self-dual. Of more interest are codes over the ring $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$. The structure of all skew constacyclic codes is completely determined. This allows us to express generators of Euclidean and Hermitian dual codes of skew cyclic and skew negacyclic codes in terms of the generators of the original codes. An illustration of all skew cyclic codes of length~2 over $\mathbb{F}_{3}+u\mathbb{F}_{3}$ and their Euclidean and Hermitian dual codes is also provided.

cs.IT↗
arXiv2009

The Gray Image of Codes over Finite Chain Rings

The results of J. F. Qiann et al. [4] on $(1-γ)$-cyclic codes over finite chain rings of nilpotency index 2 are extended to $(1-γ^e)$-cyclic codes over finite chain rings of arbitrary nilpotency index $e+1$. The Gray map is introduced for this type of rings. We prove that the Gray image of a linear $(1 - γ^{e})$-cyclic code over a finite chain ring is a distance-invariant quasi-cyclic code over its residue field. When the length of codes and the characteristic of a ring are relatively prime, the Gray images of a linear cyclic code and a linear $(1+γ^e)$-cyclic code are permutatively to quasi-cyclic codes over its residue field.

math.RA↗