Related papers: On the number of nonequivalent propelinear extende…
A perfect matching in an undirected graph $G=(V,E)$ is a set of vertex disjoint edges from $E$ that include all vertices in $V$. The perfect matching problem is to decide if $G$ has such a matching. Recently Rothvo{\ss} proved the striking…
Self-dual binary linear codes have been extensively studied and classified for length n <= 40. However, little attention has been paid to linear codes that coincide with their orthogonal complement when the underlying inner product is not…
We analyse the prequential plug-in codes relative to one-parameter exponential families M. We show that if data are sampled i.i.d. from some distribution outside M, then the redundancy of any plug-in prequential code grows at rate larger…
The minimum distance graph of a code has the codewords as vertices and edges exactly when the Hamming distance between two codewords equals the minimum distance of the code. A constructive proof for reconstructibility of an extended perfect…
A characterization of a class of optimal three-weight cyclic codes of dimension 3 over any finite field was recently presented in [10]. Shortly after this, a generalization for the sufficient numerical conditions of such characterization…
We introduce alphabet-permutation (AP) codes, a new family of error-correcting codes defined by iteratively applying random coordinate-wise permutations to a fixed initial word. A special case recovers random additive codes and random…
We initiate the probabilistic analysis of linear programming (LP) decoding of low-density parity-check (LDPC) codes. Specifically, we show that for a random LDPC code ensemble, the linear programming decoder of Feldman et al. succeeds in…
A multifold $1$-perfect code ($1$-perfect code for list decoding) in any graph is a set $C$ of vertices such that every vertex of the graph is at distance not more than $1$ from exactly $\mu$ elements of $C$. In $q$-ary Hamming graphs,…
A Z2Z4Q8-code is a non-empty subgroup of a direct product of copies of Z_2, Z_4 and Q_8 (the binary field, the ring of integers modulo 4 and the quaternion group on eight elements, respectively). Such Z2Z4Q8-codes are translation invariant…
Nearly perfect packing codes are those codes that meet the Johnson upper bound on the size of error-correcting codes. This bound is an improvement to the sphere-packing bound. A related bound for covering codes is known as the van Wee…
Exponential sums with monomials are highly related to many interesting problems in number theory and well studied by many literatures. In this paper, we consider the exponential sums with polynomials and prove a new upper bound. As an…
We construct infinite classes of almost bent and almost perfect nonlinear polynomials, which are affinely inequivalent to any sum of a power function and an affine function.
A popular method in combinatorial optimization is to express polytopes P, which may potentially have exponentially many facets, as solutions of linear programs that use few extra variables to reduce the number of constraints down to a…
We give an explicit upper bound for the number of equivalence classes of binary forms with rational integral coefficients of given degree and given discriminant, and with given splitting field. Further, we give an explicit upper bound for…
In this paper we give a detailed analysis of deterministic and randomized algorithms that enumerate any number of irreducible polynomials of degree $n$ over a finite field and their roots in the extension field in quasilinear where $N=n^2$…
We extend the theory of infinite-exponent partition relations to arbitrary linear order types, with a particular focus on the real number line. We give a complete classification of all consistent partition relations on the real line with…
A real number $x$ is absolutely normal if, for every base $b\ge 2$, every two equally long strings of digits appear with equal asymptotic frequency in the base-$b$ expansion of $x$. This paper presents an explicit algorithm that generates…
Fixed-size commutative rings are quasi-ordered such that all scalar linearly solvable networks over any given ring are also scalar linearly solvable over any higher-ordered ring. As consequences, if a network has a scalar linear solution…
In this paper, for an odd prime $p$, by extending Li et al.'s construction \cite{CL2016}, several classes of two-weight and three-weight linear codes over the finite field $\mathbb{F}_p$ are constructed from a defining set, and then their…
In this paper, a recent method to construct complementary sequence sets and complete complementary codes by Hadamard matrices is deeply studied. By taking the algebraic structure of Hadamard matrices into consideration, our main result…