Related papers: The minimum distance of parameterized codes on pro…
The distance-reduction property for a generating set, i.e., a Markov basis, of a toric ideal is a condition that ensures tight connectivity of its fibres. In this paper, we study the distance-reduction property for toric ideals of graphs…
We present conjectured candidates for the least perimeter partition of a disc into $N \le 10$ regions which take one of two possible areas. We assume that the optimal partition is connected, and therefore enumerate all three-connected…
Binary codes are constructed from incidence matrices of hypergraphs. A combinatroial description is given for the minimum distances of such codes via a combinatorial tool called ``eonv". This combinatorial approach provides a faster…
It is known that some good linear codes over a finite ring (R-linear codes) arise from interesting point constellations in certain projective geometries. For example, the expurgated Nordstrom-Robinson code, a nonlinear binary [14,6,6]-code…
We use a map to quantum error-correcting codes and a subspace projection to get lower bounds for minimal homological distances in a tensor product of two chain complexes of vector spaces over a finite field. Homology groups of such a…
In this paper, we consider minimal linear codes in a general construction of linear codes from q-ary functions. First, we give the sufficient and necessary condition for codewords to be minimal. Second, as an application, we present four…
Properties of matrix product codes over finite commutative Frobenius rings are investigated. The minimum distance of matrix product codes constructed with several types of matrices is bounded in different ways. The duals of matrix product…
The projective space of order $n$ over a finite field $\F_q$ is a set of all subspaces of the vector space $\F_q^{n}$. In this work, we consider error-correcting codes in the projective space, focusing mainly on constant dimension codes. We…
We prove that if X is any 2-regular projective scheme (in the sense of Castelnuovo-Mumford) then X is "small". This means that if L is a linear space and Y:= L\cap X is finite, then Y is "linearly independent" in the sense that the…
We introduce in this article a new method to estimate the minimum distance of codes from algebraic surfaces. This lower bound is generic, i.e. can be applied to any surface, and turns out to be ``liftable'' under finite morphisms, paving…
We show that a translation of vector V on the torus has the monotone shrinking target property if and only if the vector V is of constant type. Then, using reparametrizations of linear flows, we show that there exist area preserving real…
Let $S$ be a polynomial ring over an algebraic closed field $k$ and $ \mathfrak p =(x,y,z,w) $ a homogeneous height four prime ideal. We give a finite characterization of the degree two component of ideals primary to $\mathfrak p$, with…
An ideal projector on the space of polynomials $\mathbb{C} [\mathbf{x}]=\mathbb{C} [x_{1},\ldots ,x_{d}]$ is a projector whose kernel is an ideal in $\mathbb{C}[ \mathbf{x}]$. The question of characterization of ideal projectors that are…
We use floor decompositions of tropical curves to prove that any enumerative problem concerning conics passing through projective-linear subspaces in $\RP^n$ is maximal. That is, there exist generic configurations of real linear spaces such…
In this paper we derive geometric consequences from the presence of a long strand of linear syzygies in the minimal free resolution of a closed scheme in projective space whose homogeneous ideal is generated by quadrics. These consequences…
We examine an error-correcting coding framework in which each coded symbol is constrained to be a function of a fixed subset of the message symbols. With an eye toward distributed storage applications, we seek to design systematic codes…
Explicit bases for the subfield subcodes of projective Reed-Muller codes over the projective plane and their duals are obtained. In particular, we provide a formula for the dimension of these codes. For the general case over the projective…
The minimal free resolution of the coordinate ring of a complete intersection in projective space is a Koszul complex on a regular sequence. In the product of projective spaces $\mathbb{P}^1 \times \mathbb{P}^1$, we investigate which sets…
Reading channels where $b$-tuples of adjacent symbols are read at every step have e.g.\ applications in storage. Corresponding bounds and constructions of codes for the $b$-symbol metric, especially the pair-symbol metric where $b=2$, were…
We give local conditions at the infinite places of a number field K ensuring that the intersection of n quadrics in projective N-space over K, N >> n, satisfies weak approximation.