## Abstract

The \(p\)-ary linear code \(\mathcal {C}_{k}\!\left( n,q\right) \) is defined as the row space of the incidence matrix \(A\) of \(k\)-spaces and points of \(\textrm{PG}\!\left( n,q\right) \). It is known that if \(q\) is square, a codeword of weight \(q^k\sqrt{q}+\mathcal {O}\!\left( q^{k-1}\right) \) exists that cannot be written as a linear combination of at most \(\sqrt{q}\) rows of \(A\). Over the past few decades, researchers have put a lot of effort towards proving that any codeword of smaller weight *does* meet this property. We show that if \(q\geqslant 32\) is a composite prime power, every codeword of \(\mathcal {C}_{k}\!\left( n,q\right) \) up to weight \(\mathcal {O}\!\left( q^k\sqrt{q}\right) \) is a linear combination of at most \(\sqrt{q}\) rows of \(A\). We also generalise this result to the codes \(\mathcal {C}_{j,k}\!\left( n,q\right) \), which are defined as the \(p\)-ary row span of the incidence matrix of *k*-spaces and *j*-spaces, \(j < k\).

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## Notes

For any set \(\mathcal {S}\) of points in \(\textrm{PG}\!\left( n,q\right) \), we can define its characteristic function \(v_\mathcal {S}\) as the function that maps the points of \(\mathcal {S}\) to 1, and the other points of \(\textrm{PG}\!\left( n,q\right) \) to 0.

## References

Adriaensen S.: A note on small weight codewords of projective geometric codes and on the smallest sets of even type. SIAM J. Discret. Math.

**37**(3), 2072–2087 (2023).Adriaensen S., Denaux L.: Small weight codewords of projective geometric codes. J. Comb. Theory Ser. A

**180**, Paper No. 105395, 34 (2021).Adriaensen S., Denaux L., Storme L., Weiner Z.: Small weight code words arising from the incidence of points and hyperplanes in PG\((n, q)\). Des. Codes Cryptogr.

**88**(4), 771–788 (2020).Assmus E.F. Jr., Key J.D.: Designs and Their Codes, Volume 103 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1992).

Bagchi B.: On characterizing designs by their codes. In: Buildings, Finite Geometries and Groups, Volume 10 of Springer Proceedings in Mathematics, pp. 1–14. Springer, New York (2012).

Bagchi B.: The fourth smallest hamming weight in the code of the projective plane over \(\mathbb{Z}/p\mathbb{Z}\). arXiv:1712.07391 (2017).

Bagchi B., Inamdar S.P.: Projective geometric codes. J. Comb. Theory Ser. A

**99**(1), 128–142 (2002).Bartoli D., Denaux L.: Minimal codewords arising from the incidence of points and hyperplanes in projective spaces. Adv. Math. Commun.

**17**(1), 56–77 (2023).Blokhuis A., Brouwer A., Wilbrink H.: Hermitian unitals are code words. Discret. Math.

**97**(1–3), 63–68 (1991).Bose R.C., Burton R.C.: A characterization of flat spaces in a finite geometry and the uniqueness of the Hamming and the MacDonald codes. J. Comb. Theory

**1**, 96–104 (1966).Chouinard K.: Weight distributions of codes from finite planes. PhD Thesis, University of Virginia (2000).

De Boeck M.: Intersection problems in finite geometries. PhD Thesis, Ghent University (2014).

De Winter S., Schillewaert J., Verstraete J.: Large incidence-free sets in geometries. Electron. J. Comb.

**19**(4), Paper 24, 16 (2012).Delsarte P., Goethals J.M., MacWilliams F.J.: On generalized Reed-Muller codes and their relatives. Inf. Control

**16**, 403–442 (1970).Denaux L.: Characterising and constructing codes using finite geometries. PhD Thesis, Ghent University (2023).

Fack V., Fancsali S.L., Storme L., Van de Voorde G., Winne J.: Small weight codewords in the codes arising from Desarguesian projective planes. Des. Codes Cryptogr.

**46**(1), 25–43 (2008).Haemers W.H.: Interlacing eigenvalues and graphs. Linear Algebra Appl.

**226**(228), 593–616 (1995).Key J.D.: Hermitian varieties as codewords. Des. Codes Cryptogr.

**1**(3), 255–259 (1991).Lavrauw M., Storme L., Sziklai P., Van de Voorde G.: An empty interval in the spectrum of small weight codewords in the code from points and \(k\)-spaces of PG\((n, q)\). J. Comb. Theory Ser. A

**116**(4), 996–1001 (2009).Lavrauw M., Storme L., Van de Voorde G.: On the code generated by the incidence matrix of points and hyperplanes in PG\((n, q)\) and its dual. Des. Codes Cryptogr.

**48**(3), 231–245 (2008).Lavrauw M., Storme L., Van de Voorde G.: On the code generated by the incidence matrix of points and \(k\)-spaces in PG\((n, q)\) and its dual. Finite Fields Appl.

**14**(4), 1020–1038 (2008).Lavrauw M., Storme L., Van de Voorde G.: Linear codes from projective spaces. In: Error-Correcting Codes, Finite Geometries and Cryptography, Volume 523 of Contemporary Mathematics, pp. 185–202. American Mathematical Society, Providence, RI (2010).

McGuire G., Ward H.N.: The weight enumerator of the code of the projective plane of order \(5\). Geom. Dedicata

**73**(1), 63–77 (1998).Polverino O., Zullo F.: Codes arising from incidence matrices of points and hyperplanes in PG\((n, q)\). J. Comb. Theory Ser. A

**158**, 1–11 (2018).Szőnyi T., Weiner Z.: Stability of

*k*mod*p*multisets and small weight codewords of the code generated by the lines of PG(2, \(q\)). J. Comb. Theory Ser. A**157**, 321–333 (2018).

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Adriaensen, S., Denaux, L. Small weight codewords of projective geometric codes II.
*Des. Codes Cryptogr.* **92**, 2451–2472 (2024). https://doi.org/10.1007/s10623-024-01397-8

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DOI: https://doi.org/10.1007/s10623-024-01397-8