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Let p denote the characteristic of F q , the finite field with q elements. We prove that if q is odd then an arc of size q + 2 − t in the projective plane over F q , which is not contained in a conic, is contained in the intersection of two curves, which do not share a common component, and have degree at most t + p ⌊ log p ⁡ t ⌋ , provided a certain technical condition on t is satisfied. This implies that if q is odd then an arc of size at least q − q + q / p + 3 is contained in a conic if q is square and an arc of size at least q − q + 7 2 is contained in a conic if q is prime. This is of particular interest in the case that q is an odd square, since then there are examples of arcs, not contained in a conic, of size q − q + 1 , and it has long been conjectured that if q ≠ 9 is an odd square then any larger arc is contained in a conic. These bounds improve on previously known bounds when q is an odd square and for primes less than 1783. The previously known bounds, obtained by Segre [26] , Hirschfeld and Korchmáros [17] [18] , and Voloch [32] [33] , rely on results on the number of points on algebraic curves over finite fields, in particular the Hasse–Weil theorem and the Stöhr–Voloch theorem, and are based on Segre's idea to associate an algebraic curve in the dual plane containing the tangents to an arc. In this paper we do not rely on such theorems, but use a new approach starting from a scaled coordinate-free version of Segre's lemma of tangents. Arcs in the projective plane over F q of size q and q + 1 , q odd, were classified by Segre [25] in 1955. In this article, we complete the classification of arcs of size q − 1 and q − 2 . The main theorem also verifies the MDS conjecture for a wider range of dimensions in the case that q is an odd square. The MDS conjecture states that if 4 ⩽ k ⩽ q − 2 , a k -dimensional linear MDS code has length at most q + 1 . Here, we verify the conjecture for k ⩽ q − q / p + 2 , in the case that q is an odd square. [ABSTRACT FROM AUTHOR]

In this paper we completely characterize the words with second minimum weight in the p -ary linear code generated by the rows of the incidence matrix of points and hyperplanes of PG ( n , q ) , with q = p h and p prime, proving that they are the scalar multiples of the difference of the incidence vectors of two distinct hyperplanes of PG ( n , q ) . [ABSTRACT FROM AUTHOR]

In this paper, we prove a stability result on k mod p multisets of points in PG ( 2 , q ) , q = p h . The particular case k = 0 is used to describe small weight codewords of the code generated by the lines of PG ( 2 , q ) , as linear combination of few lines. Earlier results proved this for codewords with weight less than 2.5 q , while our result is valid until c q q . It is sharp when 27 < q square and h ≥ 4 . When q is a prime, De Boeck and Vandendriessche (see [2] ) constructed a codeword of weight 3 p − 3 that is not the linear combination of three lines. We characterise their example. [ABSTRACT FROM AUTHOR]

In this paper we point out the relationship between linear MRD-codes and various geometric objects as linearized polynomials, linear sets of P G ( n − 1 , q n ) , and generalized Segre varieties. We introduce and characterize a class of particular linear MRD-codes called F q n -linear codes with distance n − k + 1 proving that such codes are equivalent to ( n − k + 1 ) -embeddings of a canonical subgeometry. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We obtain an explicit criterion for p -ary functions to produce association schemes in terms of their Walsh spectrum. Employing this characterization, we explicitly find a correlation between p -ary bent functions and association schemes; to be more exact, we prove that a p -ary bent function induces a p -class association scheme if and only if the function is weakly regular. As applications of our main criterion, we construct many infinite families of few-class association schemes arising from p -ary functions. Furthermore, we present four classes of p -ary two-weight linear codes, which are constructed from the association schemes produced in this paper. [ABSTRACT FROM AUTHOR]

Abstract: We study -minihypers in , i.e. minihypers with the same parameters as a weighted sum of x hyperplanes. We characterize these minihypers as a nonnegative rational sum of hyperplanes and we use this characterization to extend and improve the main results of several papers which have appeared on the special case . We establish a new link with coding theory and we use this link to construct several new infinite classes of -minihypers in that cannot be written as an integer sum of hyperplanes. [Copyright &y& Elsevier]

Abstract: Let be the p-ary linear code defined by the incidence matrix of points and k-spaces in , , p prime, . In this paper, we show that there are no codewords of weight in the open interval in which implies that there are no codewords with this weight in if . In particular, for the code of points and hyperplanes of , we exclude all codewords in with weight in the open interval . This latter result implies a sharp bound on the weight of small weight codewords of , a result which was previously only known for general dimension for q prime and , with p prime, , and in the case , for , [K. Chouinard, On weight distributions of codes of planes of order 9, Ars Combin. 63 (2002) 3–13; V. Fack, Sz.L. Fancsali, L. Storme, G. Van de Voorde, J. Winne, Small weight codewords in the codes arising from Desarguesian projective planes, Des. Codes Cryptogr. 46 (2008) 25–43; M. Lavrauw, L. Storme, G. Van de Voorde, On the code generated by the incidence matrix of points and hyperplanes in and its dual, Des. Codes Cryptogr. 48 (2008) 231–245; M. Lavrauw, L. Storme, G. Van de Voorde, On the code generated by the incidence matrix of points and k-spaces in and its dual, Finite Fields Appl. 14 (2008) 1020–1038]. [Copyright &y& Elsevier]

Abstract: Given any linear code C over a finite field we show how C can be described in a transparent and geometrical way by using the associated Bruen–Silverman code. Then, specializing to the case of MDS codes we use our new approach to offer improvements to the main results currently available concerning MDS extensions of linear MDS codes. We also sharply limit the possibilities for constructing long non-linear MDS codes. Our proofs make use of the connection between the work of Rédei [L. Rédei, Lacunary Polynomials over Finite Fields, North-Holland, Amsterdam, 1973. Translated from the German by I. Földes. ] and the Rédei blocking sets that was first pointed out over thirty years ago in [A.A. Bruen, B. Levinger, A theorem on permutations of a finite field, Canad. J. Math. 25 (1973) 1060–1065]. The main results of this paper significantly strengthen those in [A. Blokhuis, A.A. Bruen, J.A. Thas, Arcs in , MDS-codes and three fundamental problems of B. Segre—Some extensions, Geom. Dedicata 35 (1–3) (1990) 1–11; A.A. Bruen, J.A. Thas, A.Blokhuis, On M.D.S. codes, arcs in with q even, and a solution of three fundamental problems of B. Segre, Invent. Math. 92 (3) (1988) 441–459]. [Copyright &y& Elsevier]

We introduce the class of partition-balanced families of codes, and show how to exploit their combinatorial invariants to obtain upper and lower bounds on the number of codes that have a prescribed property. In particular, we derive precise asymptotic estimates on the density functions of several classes of codes that are extremal with respect to minimum distance, covering radius, and maximality. The techniques developed in this paper apply to various distance functions, including the Hamming and the rank metric distances. Applications of our results show that, unlike the F q m -linear MRD codes, the F q -linear MRD codes are not dense in the family of codes of the same dimension. More precisely, we show that the density of F q -linear MRD codes in F q n × m in the set of all matrix codes of the same dimension is asymptotically at most 1/2, both as q → + ∞ and as m → + ∞. We also prove that MDS and F q m -linear MRD codes are dense in the family of maximal codes. Although there does not exist a direct analogue of the redundancy bound for the covering radius of F q -linear rank metric codes, we show that a similar bound is satisfied by a uniformly random matrix code with high probability. In particular, we prove that codes meeting this bound are dense. Finally, we compute the average weight distribution of linear codes in the rank metric, and other parameters that generalize the total weight of a linear code. [ABSTRACT FROM AUTHOR]