Minimal codes are being intensively studied in last years. n,kq-minimal linear codes are in bijection with strong blocking sets of size n in PG(k−1,q) and a lower bound for the size of strong ...blocking sets is given by (k−1)(q+1)≤n. In this note we show that all strong blocking sets of length 9 in PG(3,2) are the hyperbolic quadrics Q+(3,2).
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPUK, ZAGLJ, ZRSKP
This work investigates the structure of rank-metric codes in connection with concepts from finite geometry, most notably the q-analogues of projective systems and blocking sets. We also illustrate ...how to associate a classical Hamming-metric code to a rank-metric one, in such a way that various rank-metric properties naturally translate into the homonymous Hamming-metric notions under this correspondence. The most interesting applications of our results lie in the theory of minimal rank-metric codes, which we introduce and study from several angles. Our main contributions are bounds for the parameters of a minimal rank-metric codes, a general existence result based on a combinatorial argument, and an explicit code construction for some parameter sets that uses the notion of a scattered linear set. Throughout the paper we also show and comment on curious analogies/divergences between the theories of error-correcting codes in the rank and in the Hamming metric.
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In recent years, many connections have been made between minimal codes, a classical object in coding theory, and other remarkable structures in finite geometry and combinatorics. One of the main ...problems related to minimal codes is to give lower and upper bounds on the length m(k,q) of the shortest minimal codes of a given dimension k over the finite field Fq. It has been recently proved that m(k,q)≥(q+1)(k−1).
In this note, we prove that liminfk→∞m(k,q)k≥(q+ε(q)), where ε is an increasing function such that 1.52<ε(2)≤ε(q)≤2+12. Hence, the previously known lower bound is not tight for large enough k. We then focus on the binary case and prove some structural results on minimal codes of length 3(k−1). As a byproduct, we are able to show that, if k=5(mod8) and for other small values of k, the bound is not tight.
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Minimal rank-metric codes or, equivalently, linear cutting blocking sets are characterized in terms of the second generalized rank weight, via their connection with evasiveness properties of the ...associated
q
-system. Using this result, we provide the first construction of a family of
F
q
m
-linear MRD codes of length 2
m
that are not obtained as a direct sum of two smaller MRD codes. In addition, such a family has better parameters, since its codes possess generalized rank weights strictly larger than those of the previously known MRD codes. This shows that not all the MRD codes have the same generalized rank weights, in contrast to what happens in the Hamming metric setting.
A plane curve C⊂P2 of degree d is called blocking if every Fq-line in the plane meets C at some Fq-point. We prove that the proportion of blocking curves among those of degree d is o(1) when d≥2q−1 ...and q→∞. We also show that the same conclusion holds for smooth curves under the somewhat weaker condition d≥3p and d,q→∞. Moreover, the two events in which a random plane curve is smooth and respectively blocking are shown to be asymptotically independent. Extending a classical result on the number of Fq-roots of random polynomials, we find that the limiting distribution of the number of Fq-points in the intersection of a random plane curve and a fixed Fq-line is Poisson with mean 1. We also present an explicit formula for the proportion of blocking curves involving statistics on the number of Fq-points contained in a union of k lines for k=1,2,…,q2+q+1.
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The main purpose of this paper is to provide threshold functions for the events that a random subset of the points of a finite vector space has certain properties related to point-flat incidences. ...Specifically, we consider the events that there is an ℓ-rich m-flat with regard to a random set of points in Fqn, the event that a random set of points is an m-blocking set, and the event that there is an incidence between a random set of points and a random set of m-flats. One of our key ingredients is a stronger version of a recent result obtained by Chen and Greenhill (2021).
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A full classification (up to equivalence) of all minimal blocking sets in Desarguesian projective planes of order
≤
8 was obtained by computer. The resulting numbers of minimal blocking sets are ...tabulated according to size of the set and order of the automorphism group. For the minimal blocking sets with the larger automorphism groups explicit descriptions are given. Some of these results can also be generalised to Desarguesian projective planes of higher order.
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![All minimal [9,4]2-codes ar...](http://api.summon.serialssolutions.com/2.0.0/image/issn/XR4PS5YY4K/2666-657X/small "All minimal [9,4]2-codes are hyperbolic quadrics")
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