Related papers: Gopala-Hemachandra codes revisited
The set of all error-correcting codes C over a fixed finite alphabet F of cardinality q determines the set of code points in the unit square with coordinates (R(C), delta (C)):= (relative transmission rate, relative minimal distance). The…
We consider a uniformly rectifiable set $\Gamma \subset \mathbb R^n$ of dimension $d<n-1$. By using degenerate elliptic operators on the complement $\Omega = \mathbb R^n \setminus \Gamma$, Guy David, Svitlana Mayboroda, and the author…
In this paper we studied generalization of Hermitian function field proposed by A.Garcia and H.Stichtenoth. We calculated a Weierstrass semigroup of the point at infinity for the case q=2, r>=3. It turned out that unlike Hermitian case, we…
It is well known that a random q-ary code of rate \Omega(\epsilon^2) is list decodable up to radius (1 - 1/q - \epsilon) with list sizes on the order of 1/\epsilon^2, with probability 1 - o(1). However, until recently, a similar statement…
Generalized quasi-cyclic (GQC) codes form a natural generalization of quasi-cyclic (QC) codes. They are viewed here as mixed alphabet codes over a family of ring alphabets. Decomposing these rings into local rings by the Chinese Remainder…
This paper is concerned with the values of Harish-Chandra characters of a class of positive-depth, toral, very supercuspidal representations of $p$-adic symplectic and special orthogonal groups, near the identity element. We declare two…
A perfect code in a graph $\Gamma = (V, E)$ is a subset $C$ of $V$ that is an independent set such that every vertex in $V \setminus C$ is adjacent to exactly one vertex in $C$. A total perfect code in $\Gamma$ is a subset $C$ of $V$ such…
In this paper, harkening back to ideas of Hardy and Ramanujan, Mahler and de Bruijn, with the addition of more recent results on the Fibonacci Dirichlet series, we determine the asymptotic number of ways $p_F(n)$ to write an integer as the…
The problem to be solved in this paper is to construct a general method of proving whether a certain set is p-computable or not. The method is based on a polynomial analogue of the classical Gandhi's fixed point theorem. The classical…
In this article we prove that a class of Goppa codes whose Goppa polynomial is of the form $g(x) = x + x^q + \cdots + x^{q^{m-1}}$ where $m \geq 3$ (i.e. $g(x)$ is a trace polynomial from a field extension of degree $m \geq 3$) has a better…
Harmonic numbers $H_k=\sum_{0<j\le k}1/j (k=0,1,2,...)$ arise naturally in many fields of mathematics. In this paper we initiate the study of congruences involving both harmonic numbers and Lucas sequences. One of our three theorems is as…
General error locator polynomials are polynomials able to decode any correctable syndrome for a given linear code. Such polynomials are known to exist for all cyclic codes and for a large class of linear codes. We provide some decoding…
We adapt a construction of Guth and Lubotzky [arXiv:1310.5555] to obtain a family of quantum LDPC codes with non-vanishing rate and minimum distance scaling like $n^{0.1}$ where $n$ is the number of physical qubits. Similarly as in…
Let (F_n^{(k)})_{n\geq -(k-2)} be the k-generalized Fibonacci sequence, defined as the linear recurrence sequence whose first k terms are \(0, 0, \ldots, 0, 1\), and whose subsequent terms are determined by the sum of the preceding k terms.…
We introduce graphcodes, a novel multi-scale summary of the topological properties of a dataset that is based on the well-established theory of persistent homology. Graphcodes handle datasets that are filtered along two real-valued scale…
A generalization of the well--known Fibonacci sequence is the $k$--Fibonacci sequence with some fixed integer $k\ge 2$. The first $k$ terms of this sequence are $0,\ldots,0,1$, and each term afterwards is the sum of the preceding $k$ terms.…
Duadic codes are a class of cyclic codes that generalizes quadratic residue codes from prime to composite lengths. For every prime power q, we characterize the integers n such that over the finite field with q^2 elements there is a duadic…
We study codes with parameters of the ternary Hamming $(n=(3^m-1)/2,3^{n-m},3)$ code, i.e., ternary $1$-perfect codes. The rank of the code is defined to be the dimension of its affine span. We characterize ternary $1$-perfect codes of rank…
Quantum error-correcting codes aim to protect information in quantum systems to enable fault-tolerant quantum computations. The most prevalent method, stabilizer codes, has been well developed for many varieties of systems, however, largely…
A beautiful theorem of Zeckendorf states that every integer can be written uniquely as a sum of non-consecutive Fibonacci numbers $\{F_n\}_{n=1}^{\infty}$. Lekkerkerker proved that the average number of summands for integers in $[F_n,…