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Let $C$ be a linear code of length $n$ and dimension $k$ over the finite field $\mathbb{F}_{q^m}$. The trace code $\mathrm{Tr}(C)$ is a linear code of the same length $n$ over the subfield $\mathbb{F}_q$. The obvious upper bound for the dimension of the trace code over $\mathbb{F}_q$ is $mk$. If equality holds, then we say that $C$ has maximum trace dimension. The problem of finding the true dimension of trace codes and their duals is relevant for the size of the public key of various code-based cryptographic protocols. Let $C_{\mathbf{a}}$ denote the code obtained from $C$ and a multiplier vector $\mathbf{a}\in (\mathbb{F}_{q^m})^n$. In this paper, we give a lower bound for the probability that a random multiplier vector produces a code $C_{\mathbf{a}}$ of maximum trace dimension. We give an interpretation of the bound for the class of algebraic geometry codes in terms of the degree of the defining divisor. The bound explains the experimental fact that random alternant codes have minimal dimension. Our bound holds whenever $n\geq m(k+h)$, where $h\geq 0$ is the Singleton defect of $C$. For the extremal case $n=m(h+k)$, numerical experiments reveal a closed connection between the probability of having maximum trace dimension and the probability that a random matrix has full rank.

Let $k\ge 2$ be an integer and let $A$ be a set of nonnegative integers. For a $k$-tuple of positive integers $\underline{\lambda} = (\lambda_{1}, \dots{} ,\lambda_{k})$ with $1 \le \lambda_{1} < \lambda_{2} < \dots{} < \lambda_{k}$, we define the additive representation function $R_{A,\underline{\lambda}}(n) = |\{(a_{1}, \dots{} ,a_{k})\in A^{k}: \lambda_{1}a_{1} + \dots{} + \lambda_{k}a_{k} = n\}|$. For $k = 2$, Moser constructed a set $A$ of nonnegative integers such that $R_{A,\underline{\lambda}}(n) = 1$ holds for every nonnegative integer $n$. In this paper we characterize all the $k$-tuples $\underline{\lambda}$ and the sets $A$ of nonnegative integers with $R_{A,\underline{\lambda}}(n) = 1$ for every integer $n\ge 0$.

Let $k\ge 2$ be an integer and let $A$ be a set of nonnegative integers. The representation function $R_{A,k}(n)$ for the set $A$ is the number of representations of a nonnegative integer $n$ as the sum of $k$ terms from $A$. Let $A(n)$ denote the counting function of $A$.Bell and Shallit recently gave a counterexample for a conjecture of Dombi and proved that if $A(n)=o(n^{\frac{k-2}{k}-\epsilon})$ for some $\epsilon>0$, then $R_{\mathbb{N}\setminus A,k}(n)$ is eventually strictly increasing. In this paper, we improve this result to $A(n)=O(n^{\frac{k-2}{k-1}})$. We also give an example to show that this bound is best possible., Comment: 11 pages

Let $h,k \ge 2$ be integers. A set $A$ of positive integers is called asymptotic basis of order $k$ if every large enough positive integer can be written as the sum of $k$ terms from $A$. A set of positive integers $A$ is said to be a $B_{h}[g]$-set if every positive integer can be written as the sum of $h$ terms from $A$ at most $g$ different ways. In this paper we prove the existence of $B_{h}[1]$ sets which are asymptotic bases of order $2h$ by using probabilistic methods., Comment: arXiv admin note: text overlap with arXiv:2103.10349

Let $k \ge 2$ be an integer. We say a set $A$ of positive integers is an asymptotic basis of order $k$ if every large enough positive integer can be represented as the sum of $k$ terms from $A$. A set of positive integers $A$ is called Sidon set if all the two terms sums formed by the elements of $A$ are different. Many years ago P. Erd\H{o}s, A. S\'ark\"ozy and V. T. S\'os asked whether there exists a Sidon set which is asymptotic basis of order $3$. In this paper we prove the existence of a Sidon set $A$ with positive lower density of the three fold sumset $A + A + A$ by using probabilistic methods., Comment: arXiv admin note: text overlap with arXiv:1304.5749

Two infinite sets $A$ and $B$ of nonnegative integers are called additive complements if their sumset contains every nonnegative integer. In 1964, Danzer constructed infinite additive complements $A$ and $B$ with $A(x)B(x) = (1 + o(1))x$ as $x \rightarrow \infty$, where $A(x)$ and $B(x)$ denote the counting function of the sets $A$ and $B$, respectively. In this paper we solve a problem of Chen and Fang by extending the construction of Danzer.

Let $h,k \ge 2$ be integers. We say a set $A$ of positive integers is an asymptotic basis of order $k$ if every large enough positive integer can be represented as the sum of $k$ terms from $A$. A set of positive integers $A$ is called $B_{h}[g]$ set if all positive integers can be represented as the sum of $h$ terms from $A$ at most $g$ times. In this paper we prove the existence of $B_{h}[1]$ sets which are asymptotic bases of order $2h+1$ by using probabilistic methods.

We propose a new identification system based on algorithmic problems related to computing isomorphisms between central simple algebras. We design a statistical zero knowledge protocol which relies on the hardness of computing isomorphisms between orders in division algebras which generalizes a protocol by Hartung and Schnorr, which relies on the hardness of integral equivalence of quadratic forms., Comment: accepted to Applicable Algebra in Engineering, Communication and Computing

Let $A = \{a_{1},a_{2},\dots{}\}$ $(a_{1} < a_{2} < \dots{})$ be an infinite sequence of nonnegative integers, and let $R_{A,2}(n)$ denote the number of solutions of $a_{x}+a_{y}=n$ $(a_{x},a_{y}\in A)$. P. Erd\H{o}s, A. S\'ark\"ozy and V. T. S\'os proved that if $\lim_{N\to\infty}\frac{B(A,N)}{\sqrt{N}}=+\infty$ then $|\Delta_{1}(R_{A,2}(n))|$ cannot be bounded, where $B(A,N)$ denotes the number of blocks formed by consecutive integers in $A$ up to $N$ and $\Delta_{l}$ denotes the $l$-th difference. Their result was extended to $\Delta_{l}(R_{A,2}(n))$ for any fixed $l\ge2$. In this paper we give further generalizations of this problem.

Let $\mathbb{N}$ denote the set of all nonnegative integers. Let $k\ge 3$ be an integer and $A_{0} = \{a_{1}, \dots{}, a_{t}\}$ $(a_{1} < \ldots< a_{t})$ be a nonnegative set which does not contain an arithmetic progression of length $k$. We denote $A = \{a_{1}, a_{2}, \dots{}\}$ defined by the following greedy algorithm: if $l \ge t$ and $a_{1}, \dots{}, a_{l}$ have already been defined, then $a_{l+1}$ is the smallest integer $a > a_{l}$ such that $\{a_{1}, \dots{}, a_{l}\} \cup \{a\}$ also does not contain a $k$-term arithmetic progression. This sequence $A$ is called the Stanley sequence of order $k$ generated by $A_{0}$. In this paper, we prove some results about various generalizations of the Stanley sequence.

Let $\mathbb{Z}^{+}$ be the set of positive integers. Let $C_{k}$ denote all subsets of $\mathbb{Z}^{+}$ such that neither of them contains $k + 1$ pairwise coprime integers and $C_k(n)=C_k\cap \{1,2,\ldots,n\}$. Let $f(n, k) = \text{max}_{A \in C_{k}(n)}|A|$, where $|A|$ denotes the number of elements of the set $A$. Let $E_k(n)$ be the set of positive integers not exceeding $n$ which are divisible by at least one of the primes $p_{1}, \dots{}, p_{k}$, where $p_{i}$ denote the $i$th prime number. In 1962, Erd\H{o}s conjectured that $f(n, k) = |E(n,k)|$ for every $n \ge p_{k}$. Recently Chen and Zhou proved some results about this conjecture. In this paper we solve an open problem of Chen and Zhou and prove several related results about the conjecture., Comment: 19 pages

Let $C,W\subseteq \mathbb{Z}$. If $C+W=\mathbb{Z}$, then the set $C$ is called an additive complement to $W$ in $\mathbb{Z}$. If no proper subset of $C$ is an additive complement to $W$, then $C$ is called a minimal additive complement. Let $X\subseteq \mathbb{N}$. If there exists a positive integer $T$ such that $x+T\in X$ for all sufficiently large integers $x\in X$, then we call $X$ eventually periodic. In this paper, we study the existence of a minimal complement to $W$ when $W$ is eventually periodic or not. This partially answers a problem of Nathanson., Comment: 13 pages

For a set of nonnegative integers $A$, denote by $R_{A}(n)$ the number of unordered representations of the integer $n$ as the sum of two different terms from $A$. In this paper we partially describe the structure of the sets, which have coinciding representation functions.

For a set of nonnegative integers $S$ let $R_{S}(n)$ denote the number of unordered representations of the integer $n$ as the sum of two different terms from $S$. In this paper we focus on partitions of the natural numbers into two sets affording identical representation functions. We solve a recent problem of Lev and Chen., Comment: Theorem 7. was cancelled from the previous version

Let $A$ and $B$ be sets of nonnegative integers. For a positive integer $n$ let $R_{A}(n)$ denote the number of representations of $n$ as the sum of two terms from $A$. Let $\displaystyle s_{A}(x) = \max_{n \le x}R_{A}(n)$ and $\displaystyle d_{A,B}(x) = \max_{\hbox{t: $a_{t} \le x$ or $b_{t} \le x$}}|a_{t} - b_{t}|$. In this paper we study the connection between $s_{A}(x)$, $s_{B}(x)$ and $d_{A,B}(x)$. We improve a result of Haddad and Helou about the Erd\H{o}s - Tur\'an conjecture.

We say the sets of nonnegative integers A and B are additive complements if their sum contains all sufficiently large integers. In this paper we prove a conjecture of Chen and Fang about additive complement of a finite set.

In this paper we prove the existence of Sidon sets which are asymptotic bases of order 4 by using probabilistic methods.

The Single Cut or Join (SCJ) operation on genomes, generalizing chromosome evolution by fusions and fissions, is the computationally simplest known model of genome rearrangement. While most genome rearrangement problems are already hard when comparing three genomes, it is possible to compute in polynomial time a most parsimonious SCJ scenario for an arbitrary number of genomes related by a binary phylogenetic tree. Here we consider the problems of sampling and counting the most parsimonious SCJ scenarios. We show that both the sampling and counting problems are easy for two genomes, and we relate SCJ scenarios to alternating permutations. However, for an arbitrary number of genomes related by a binary phylogenetic tree, the counting and sampling problems become hard. We prove that if a Fully Polynomial Randomized Approximation Scheme or a Fully Polynomial Almost Uniform Sampler exist for the most parsimonious SCJ scenario, then RP = NP. The proof has a wider scope than genome rearrangements: the same result holds for parsimonious evolutionary scenarios on any set of discrete characters.

With the current burst of network theory (especially in connection with social and biological networks) there is a renewed interest on realizations of given degree sequences. In this paper we propose an essentially new degree sequence problem: we want to find graphical realizations of a given degree sequence on labeled vertices, where certain would-be edges are {\em forbidden}. Then we want to sample uniformly and efficiently all these possible realizations. (This problem can be considered as a special case of Tutte's $f$-factor problem, however it has a favorable sampling speed.) We solve this {\em restricted degree sequence} (or RDS for short) problem completely if the forbidden edges form a bipartite graph, which consist of the union of a (not necessarily maximal) 1-factor and a (possible empty) star. Then we show how one can sample the space of all realizations of these RDSs uniformly and efficiently when the degree sequence describes a {\em half-regular} bipartite graph. Our result contains, as special cases, the well-known result of Kannan, Tetali and Vempala on sampling regular bipartite graphs and a recent result of Greenhill on sampling regular directed graphs (so it also provides new proofs of them). The RDS problem descried above is self-reducible, therefore our {\em fully polynomial almost uniform sampler} (a.k.a. FPAUS) on the space of all realizations also provides a {\em fully polynomial randomized approximation scheme} (a.k.a. FPRAS) for approximate counting of all realizations.

Erd\H os and R\'{e}nyi claimed and Vu proved that for all $h \ge 2$ and for all $\epsilon > 0$, there exists $g = g_h(\epsilon)$ and a sequence of integers $A$ such that the number of ordered representations of any number as a sum of $h$ elements of $A$ is bounded by $g$, and such that $|A \cap [1,x]| \gg x^{1/h - \epsilon}$. We give two new proofs of this result. The first one consists of an explicit construction of such a sequence. The second one is probabilistic and shows the existence of such a $g$ that satisfies $g_h(\epsilon) \ll \epsilon^{-1}$, improving the bound $g_h(\epsilon) \ll \epsilon^{-h+1}$ obtained by Vu. Finally we use the "alteration method" to get a better bound for $g_3(\epsilon)$, obtaining a more precise estimate for the growth of $B_3[g]$ sequences., Comment: 12 pages, no figures