András Gács

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We show that if a linear code admits an extension, then it necessarily admits a linear extension. There are many linear codes that are known to admit no linear extensions. Our result implies that these codes are in fact maximal. We are able to characterize maximal linear (n, k, d)q-codes as complete (weighted) (n, n − d)-arcs in PG(k−1, q). At the same time(More)
In this paper we outline a construction method which has been used for minimal blocking sets in PG(2, q) and maximal partial line spreads in PG(n, q) and which must have a lot of more applications. We also give a survey on what is known about the spectrum of sizes of maximal partial line spreads in PG(n, q). At the end we list some more elaborate random(More)
A three-dimensional analogue of the classical direction problem is proposed and an asymptotically sharp bound for the number of directions determined by a nonplanar set in AG(3, p), p prime, is proved. Using the terminology of permutation polynomials the main result states that if there are more than (2dp−1 6 e+1)(p+2d p−1 6 e)/2 ≈ 2p/9 pairs (a, b) ∈ Fp(More)
Let f be a function from a finite field Fp with a prime number p of elements, to Fp. In this article we consider those functions f(X) for which there is a positive integer n > 2 √ p− 1− 11 4 with the property that f(X) , when considered as an element of Fp[X]/(X −X), has degree at most p− 2− n + i, for all i = 1, . . . , n. We prove that every line is(More)