Designs, Codes and Cryptography

, Volume 87, Issue 4, pp 841–854 | Cite as

Linear codes close to the Griesmer bound and the related geometric structures

  • Assia RoussevaEmail author
  • Ivan Landjev
Part of the following topical collections:
  1. Special Issue: Finite Geometries


In this paper, we study the behavior of the function \(t_q(k)\) defined as the maximal deviation from the Griesmer bound of the optimal length of a linear code with a fixed dimension k:
$$\begin{aligned} t_q(k)=\max _d(n_q(k,d)-g_q(k,d)), \end{aligned}$$
where the maximum is taken over all minimum distances d. Here \(n_q(k,d)\) is the shortest length of a q-ary linear code of dimension k and minimum distance d, \(g_q(k,d)\) is the Griesmer bound for a code of dimension k and minimum distance d. We give two equivalent geometric versions of this problem in terms of arcs and minihypers. We prove that \(t_q(k)\rightarrow \infty \) when \(k\rightarrow \infty \) which implies that the problem is non-trivial. We prove upper bounds on the function \(t_q(k)\). For codes of even dimension k we show that \(t_q(k)\le 2(q^{k/2}-1)/(q-1)-(k+q-1)\) which implies that \(t_q(k)\in O(q^{k/2})\) for all k. For three-dimensional codes and q even we prove the stronger estimate \(t_q(3)\le \log q-1\), as well as \(t_q(3)\le \sqrt{q}-1\) for the case q square.


Griesmer bound Optimal linear codes Arcs Minihypers 

Mathematics Subject Classification

94B65 51E21 51E22 94B05 94B27 51E15 51E23 



This result was partially supported by the Research Scientific Fund of Sofia University “St. Kl. Ohridski” under Contract No. 80-10-55/19.04.2017.


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Authors and Affiliations

  1. 1.Sofia UniversitySofiaBulgaria
  2. 2.New Bulgarian UniversitySofiaBulgaria
  3. 3.Institute of Mathematics and InformaticsBulgarian Academy of SciencesSofiaBulgaria

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