Related papers: On the code generated by the incidence matrix of p…

In this paper, we study the p-ary linear code C(PG(n, q)), q = p^h, p prime, h >= 1, generated by the incidence matrix of points and hyperplanes of a Desarguesian projective space PG(n, q), and its dual code. We link the codewords of small…

We investigate small weight code words of the $p$-ary linear code $\mathcal C_{j,k}(n,q)$ generated by the incidence matrix of $k$-spaces and $j$-spaces of PG$(n,q)$ and its dual, with $q$ a prime power and $0 \leq j < k < n$. Firstly, we…

In this paper we completely characterize the words with second minimum weight in the $p-$ary linear code generated by the rows of the incidence matrix of points and hyperplanes of $PG(n,q)$, with $q=p^h$ and $p$ prime, proving that they are…

Let Ck(n, q) be the p-ary linear code defined by the incidence matrix of points and k-spaces in PG(n, q), q = p^h, p prime, h >= 1. In this pa- per, we show that there are no codewords of weight in the open interval ] q^{k+1}-1/q-1, 2q^k[…

The $p$-ary linear code $\mathcal C_{k}(n,q)$ is defined as the row space of the incidence matrix $A$ of $k$-spaces and points of $\text{PG}(n,q)$. It is known that if $q$ is square, a codeword of weight $q^k\sqrt{q}+\mathcal O \left(…

In this paper, we study the codes $\mathcal C_k(n,q)$ arising from the incidence of points and $k$-spaces in $\text{PG}(n,q)$ over the field $\mathbb F_p$, with $q = p^h$, $p$ prime. We classify all codewords of minimum weight of the dual…

Over the past few years, the codes $\mathcal{C}_{n-1}(n,q)$ arising from the incidence of points and hyperplanes in the projective space $\text{PG}(n,q)$ attracted a lot of attention. In particular, small weight codewords of…

The $p$-ary code associated with the incidence structure of points and $t$-spaces in a projective space $\mathrm{PG}(m,q)$, where $q=p^h$, is the $\mathbb{F}_p$-subspace generated by the incidence vectors of the blocks of this design. The…

Let $C_{n-1}(n,q)$ be the code arising from the incidence of points and hyperplanes in the Desarguesian projective space PG($n,q$). Recently, Polverino and Zullo proved that within this code, all non-zero code words of weight at most…

In this paper, we prove a stability result on k mod p multisets of points in PG(2,q), q = p^h. The particular case k=0 is used to describe small weight codewords of the code generated by the lines of PG(2, q), as linear combination of few…

A small minimal k-blocking set B in PG(n, q), q = pt, p prime, is a set of less than 3(qk + 1)/2 points in PG(n, q), such that every (n - k)-dimensional space contains at least one point of B and such that no proper subset of B satisfies…

In this paper, we show that a small minimal k-blocking set in PG(n, q3), q = p^h, h >= 1, p prime, p >=7, intersecting every (n-k)-space in 1 (mod q) points, is linear. As a corollary, this result shows that all small minimal k-blocking…

In this paper we consider two pointsets in $\mathrm{PG}(2,q^n)$ arising from a linear set $L$ of rank $n$ contained in a line of $\mathrm{PG}(2,q^n)$: the first one is a linear blocking set of R\'edei type, the second one extends the…

Let $\mathcal{O}$ be a conic in the classical projective plane $PG(2,q)$, where $q$ is an odd prime power. With respect to $\mathcal{O}$, the lines of $PG(2,q)$ are classified as passant, tangent, and secant lines, and the points of…

For which positive integers $n,k,r$ does there exist a linear $[n,k]$ code $C$ over $\mathbb{F}_q$ with all codeword weights divisible by $q^r$ and such that the columns of a generating matrix of $C$ are projectively distinct? The…

In [9], the codewords of small weight in the dual code of the code of points and lines of Q(4, q) are characterised. Inspired by this result, using geometrical arguments, we characterise the codewords of small weight in the dual code of the…

Linear codes with a few weights are an important class of codes in coding theory and have attracted a lot of attention. In this paper, we present several constructions of $q$-ary linear codes with two or three weights from vectorial…

In this paper, we consider the unit graph $G(\mathbb{Z}_{n})$, where $n=p_{1}^{n_{1}} \text{ or } p_{1}^{n_{1}}p_{2}^{n_{2}} \text{ or } p_{1}^{n_{1}}p_{2}^{n_{2}}p_{3}^{n_{3}}$ and $p_{1}, p_{2}, p_{3}$ are distinct primes. For any prime…

In this paper, we prove that the smallest even sets in ${\rm PG}(n,q)$, i.e. sets that intersect every line in an even number of points, are cylinders with a hyperoval as base. This fits into a more general study of dual projective…

Let A be the incidence matrix of passant lines and internal points with respect to a conic in PG(2, q), where q is an odd prime power. In this article, we study both geometric and algebraic properties of the column null space L of A over…