Related papers: On perfect codes that do not contain Preparata-lik…
We describe a new construction of a subset of P^4 with no four points on a plane over any finite field of order q in which 3 is not a square. This set has size 2q + 1, is maximal with respect to inclusion, and is the largest known such set.
In this paper, we show that an extremal Type II $\ZZ_{2k}$-code of length $n$ dose not exist for all sufficiently large $n$ when $k=2,3,4,5,6$.
It is conjectured by Golomb and Welch around half a century ago that there is no perfect Lee codes $C$ of packing radius $r$ in $\mathbb{Z}^{n}$ for $r\geq2$ and $n\geq 3$. Recently, Leung and the second author proved this conjecture for…
We prove that there are no Wieferich's primes $q=2p+1$ where $p \equiv 3 \pmod{4}$ is a prime number
In this work, we compute the perfect forms for all imaginary quadratic fields of absolute discriminant up to $5000$ and study the number and types of the polytopes that arise. We prove a bound on the combinatorial types of polytopes that…
We find all positive integer solutions in $x, y$ and $n$ of $x^2+19^{2k+1}=4y^{n}$ for any non-negative integer $k$.
In this paper, we proved that there are infinite cube--free numbers of the form $[n^c]$ for any fixed real number $1<c<11/6$.
This paper makes the following conjecture: For every prime $p$ there exists a positive integer $x$ with $\left\lceil \frac{p}{4} \right\rceil \leq x \leq \left\lceil \frac{p}{2} \right\rceil$ and a positive divisor $d|x^2$ so that either:…
Much work has been done to identify which binary codes can be represented by collections of open convex or closed convex sets. While not all binary codes can be realized by such sets, here we prove that every binary code can be realized by…
The well known binary and decimal representations of the integers, and other similar number systems, admit many generalisations. Here, we investigate whether still every integer could have a finite expansion on a given integer base b, when…
Each linear code can be described by a code ideal given as the sum of a toric ideal and a non-prime ideal. In this way, several concepts from the theory of toric ideals can be translated into the setting of code ideals. It will be shown…
A rational positive-definite quadratic form is perfect if it can be reconstructed from the knowledge of its minimal nonzero value m and the finite set of integral vectors v such that f(v) = m. This concept was introduced by Voronoi and…
We revisit the topic of power-free morphisms, focusing on the properties of the class of complementary morphisms. Such morphisms are defined over a $2$-letter alphabet, and map the letters 0 and 1 to complementary words. We prove that every…
We present a linear functional calculus with both the safety guarantees expressible with linear types and the rich language of combinators and composition provided by functional programming. Unlike previous combinations of linear typing and…
We prove that any ordered field can be extended to one for which every decreasing sequence of bounded closed intervals, of any length, has a nonempty intersection; equivalently, there are no Dedekind cuts with equal cofinality from both…
In this paper, based on the nonbinary graph state, we present a systematic way of constructing good non-binary quantum codes, both additive and nonadditive, for systems with integer dimensions. With the help of computer search, which…
A perfect code in a graph $\Gamma = (V, E)$ is a subset $C$ of $V$ such that no two vertices in $C$ are adjacent and every vertex in $V \setminus C$ is adjacent to exactly one vertex in $C$. A total perfect code in $\Gamma$ is a subset $C$…
We study the structure of the language of binary cube-free words. Namely, we are interested in the cube-free words that cannot be infinitely extended preserving cube-freeness. We show the existence of such words with arbitrarily long finite…
For some extremal (optimal) odd unimodular lattices L in dimensions n=12,16,20,32,36,40 and 44, we determine all positive integers k such that L contains a k-frame. This result yields the existence of an extremal Type I Zk-code of lengths…
Motivated by DNA storage in living organisms, and by known biological mutation processes, we study the reverse-complement string-duplication system. We fully classify the conditions under which the system has full expressiveness, for all…