arXiv subjects

Tai Do Duc

Publications and source records attributed to Tai Do Duc.

8 recordsLinked to original sources

arXiv2020

Symmetric uncoded caching schemes with low subpacketization levels

Caching is a commonly used technique in content-delivery networks which aims to deliver information from hosting servers to users in the most efficient way. In 2014, Maddah-Ali and Niessen formulated caching into a formal information theoretic problem and it has gained a lot of attention since then. It is known that the caching schemes proposed by Ali-Niesen and Yu et. al. are optimal, that is, they require the least number of transmissions from the server to satisfy all users' demands. However for these schemes to work, each file needs to be partitioned into $F^*$ subfiles ($F^*$ is called the subpacketization level of files) with $F^*$ growing exponentially in the number $K$ of users. As a result, it is problematic to apply these schemes in practical situations, where $K$ tends to be very large. There rise the following questions: (1) are there optimal schemes in which each file is partitioned into $F$ subfiles, where $F$ is not exponential, say polynomial for example, in $K$? (2) if the answer to this question is no, is there a near-optimal scheme, a scheme which is as asymptotically good as the one in \cite{ali1,yu}, with $F$ polynomial in $K$? Both these questions are open. Our main contribution in this paper is to provide answers to above questions. Firstly, we prove that under some mild restriction on user's cache rate, there are no optimal schemes with $F$ smaller than $F^*$. Moreover, we give necessary and sufficient conditions for the existence of optimal schemes in this case. Secondly, we provide an affirmative answer to the second question raised above by an explicit construction and a detailed performance analysis.

cs.IT↗

arXiv2019

New Constructions of Group-Invariant Butson Hadamard Matrices

Let $G$ be a finite group and let $h$ be a positive integer. A $\text{BH}(G,h)$ matrix is a $G$-invariant $|G|\times |G|$ matrix $H$ whose entries are complex $h$th roots of unity such that $HH^*=|G|I_{|G|}$, where $H^*$ denotes the complex conjugate transpose of $H$, and $I_{|G|}$ is the identity matrix of order $|G|$. In this paper, we give three new constructions of $\text{BH}(G,h)$ matrices. The first construction is the first known family of $\text{BH}(G,h)$ matrices in which $G$ does not need to be abelian. The second and the third constructions are two families of $\text{BH}(G,h)$ matrices in which $G$ is a finite local ring.

math.CO↗

arXiv2019

Explicit Constructions of Two-Dimensional Reed-Solomon Codes in High Insertion and Deletion Noise Regime

Insertion and deletion (insdel for short) errors are synchronization errors in communication systems caused by the loss of positional information in the message. Reed-Solomon codes have gained a lot of interest due to its encoding simplicity, well structuredness and list-decoding capability in the classical setting. This interest also translates to the insdel metric setting, as the Guruswami-Sudan decoding algorithm can be utilized to provide a deletion correcting algorithm in the insdel metric. Nevertheless, there have been few studies on the insdel error-correcting capability of Reed-Solomon codes. Our main contributions in this paper are explicit constructions of two families of 2-dimensional Reed-Solomon codes with insdel error-correcting capabilities asymptotically reaching those provided by the Singleton bound. The first construction gives a family of Reed-Solomon codes with insdel error-correcting capability asymptotic to its length. The second construction provides a family of Reed Solomon codes with an exact insdel error-correcting capability up to its length. Both our constructions improve the previously known construction of 2-dimensional Reed-Solomon codes whose insdel error-correcting capability is only logarithmic on the code length.

cs.IT↗

arXiv2019

Necessary Conditions for the Existence of Group-Invariant Butson Matrices and a New Family of Perfect Arrays

Let $G$ be a finite abelian group and let $\exp(G)$ denote the least common multiple of the orders of all elements of $G$. A $BH(G,h)$ matrix is a $G$-invariant $|G|\times |G|$ matrix $H$ whose entries are complex $h$th roots of unity such that $HH^*=|G|I_{|G|}$. In this paper, we study the relation between $G$ and $h$ so that a $BH(G,h)$ matrix exists. We will only focus on $BH(\mathbb{Z}_n,h)$ matrices and $BH(G,2p^b)$ matrices, where $p$ is an odd prime. By our results, there are $2687$ open cases left for the existence of $BH(\mathbb{Z}_n,h)$ matrices in which $1\leq n,h \leq 100$. In the last section, we show that $BH(\mathbb{Z}_n,h)$ matrices can be used to construct a new family of perfect polyphase arrays.

math.CO↗

arXiv2019

Bilinear Forms on Finite Abelian Groups and Group-Invariant Butson Hadamard Matrices

Let $K$ be a finite abelian group and let $\exp(K)$ denote the least common multiple of the orders of the elements of $K$. A $BH(K,h)$ matrix is a $K$-invariant $|K|\times |K|$ matrix $H$ whose entries are complex $h$th roots of unity such that $HH^*=|K|I$, where $H^*$ denotes the complex conjugate transpose of $H$, and $I$ is the identity matrix of order $|K|$. Let $ν_p(x)$ denote the $p$-adic valuation of the integer $x$. Using bilinear forms on $K$, we show that a $BH(K,h)$ exists whenever (i) $ν_p(h) \geq \lceil ν_p(\exp(K))/2 \rceil$ for every prime divisor $p$ of $|K|$ and (ii) $ν_2(h) \ge 2$ if $ν_2(|K|)$ is odd and $K$ has a direct factor $\mathbb{Z}_2$. Employing the field descent method, we prove that these conditions are necessary for the existence of a $BH(K,h)$ matrix in the case where $K$ is cyclic of prime power order.

math.CO↗

arXiv2019

Upper Bounds for Cyclotomic Numbers

Let $q$ be a power of a prime $p$, let $k$ be a nontrivial divisor of $q-1$ and write $e=(q-1)/k$. We study upper bounds for cyclotomic numbers $(a,b)$ of order $e$ over the finite field $\mathbb{F}_q$. A general result of our study is that $(a,b)\leq 3$ for all $a,b \in \mathbb{Z}$ if $p> (\sqrt{14})^{k/ord_k(p)}$. More conclusive results will be obtained through separate investigation of the five types of cyclotomic numbers: $(0,0), (0,a), (a,0), (a,a)$ and $(a,b)$, where $a\neq b$ and $a,b \in \{1,\dots,e-1\}$. The main idea we use is to transform equations over $\mathbb{F}_q$ into equations over the field of complex numbers on which we have more information. A major tool for the improvements we obtain over known results is new upper bounds on the norm of cyclotomic integers.

math.NT↗

arXiv2019

Non-projective cyclic codes whose check polynomial contains two zeros

Let $n\geq 3$ be a positive integer and let $\mathbb{F}_{q^k}$ be the splitting field of $x^n-1$. By $γ$ we denote a primitive element of $\mathbb{F}_{q^k}$. Let $C$ be a cyclic code of length $n$ whose check polynomial contains two zeros $γ^d$ and $γ^{d+D}$, where $de \mid (q-1)$, $e>1$ and $D=(q^k-1)/e$. This family of cyclic codes is not projective. Many authors have studied the weight distribution of these codes for certain parameters. In this paper, we prove that these codes are never two-weight codes. This result would strengthen a conjecture by Vega which states that all two-weight cyclic codes are the "known" ones.

math.CO↗

arXiv2019

Unique Differences in Symmetric Subsets of $\mathbb{F}_p$

Let $p$ be a prime and let $A$ be a subset of $\mathbb{F}_p$ with $A=-A$ and $|A\setminus\{0\}| \leq 2\log_3(p)$. Then there is an element of $\mathbb{F}_p$ which has a unique representation as a difference of two elements of $A$.

math.CO↗