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Short Minimal Codes and Covering Codes via Strong Blocking Sets in Projective Spaces | IEEE Journals & Magazine | IEEE Xplore

Short Minimal Codes and Covering Codes via Strong Blocking Sets in Projective Spaces


Abstract:

Minimal linear codes are in one-to-one correspondence with special types of blocking sets of projective spaces over a finite field, which are called strong or cutting blo...Show More

Abstract:

Minimal linear codes are in one-to-one correspondence with special types of blocking sets of projective spaces over a finite field, which are called strong or cutting blocking sets. Minimal linear codes have been studied since decades but their tight connection with cutting blocking sets of finite projective spaces was unfolded only in the past few years, and it has not been fully exploited yet. In this paper we apply finite geometric and probabilistic arguments to contribute to the field of minimal codes. We prove an upper bound on the minimal length of minimal codes of dimension k over the q -element Galois field which is linear in both q and k , hence improve the previous superlinear bounds. This result determines the minimal length up to a small constant factor. We also improve the lower and upper bounds on the size of so called higgledy-piggledy line sets in projective spaces and apply these results to present improved bounds on the size of covering codes and saturating sets in projective spaces as well.
Published in: IEEE Transactions on Information Theory ( Volume: 68, Issue: 2, February 2022)
Page(s): 881 - 890
Date of Publication: 28 October 2021

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