In this paper, we determine the dual codes of cyclic codes of length over , where p is a prime number and . Also, we improve and give correction of the results stated by B. Kim and J. Lee (2020) in [11]. Finally, we provide some examples of optimal and near-MDS cyclic codes of length over and compute dual of them.
Stability results play an important role in Galois geometries. The famous resultant method, developed by Szőnyi and Weiner [12], [11], became very fruitful and resulted in many stability theorems in the last two decades. This method is based on some bivariate polynomials associated to point sets. In this paper we generalize the method for the multidimensional case and show some new applications. We build up the multivariate polynomial machinery and apply it for Rédei polynomials. We can prove a high dimensional analogue of the result of Szőnyi-Weiner [9], concerning the number of hyperplanes being skew to a point set of the space. We prove general results on “partial blocking sets”, using the tools we have developed.
In this paper, we revisit the algorithm for computing chains of -isogenies between products of elliptic curves via theta coordinates proposed by Dartois et al. For each fundamental block of this algorithm, we provide an explicit inversion-free version. Besides, we exploit the technique of x-only ladder to speed up the computation of gluing isogeny. Finally, we present a mixed optimal strategy, which combines the inversion-elimination tool with the original methods together to execute a chain of -isogenies.
We make a cost analysis and present a concrete comparison between ours and the previously known methods for inversion elimination. Furthermore, we implement the mixed optimal strategy for benchmark. The results show that when computing -isogeny chains with lengths of 126, 208 and 632, compared to Dartois, Maino, Pope and Robert's latest implementation, utilizing our techniques can reduce 9.7%, 9.5% and 9.6% multiplications over the base field , respectively. Therefore, even for the updated version that employs their inversion-free algorithms, our tools still possess an advantage.
A complete partition of a graph G is a partition of the vertex set such that there is at least one edge between any two parts. The largest r such that G has a complete partition into r parts, each of which is an independent set, is the achromatic number of G. We determine the achromatic number of polarity graphs of biaffine planes coming from generalized polygons. Our colorings of a family of unitary polarity graphs are used to solve a problem of Axenovich and Martin on complete partitions of -free graphs. Furthermore, these colorings prove that there are sequences of graphs which are optimally complete and have unbounded degree, a problem that had been studied for the sequence of hypercubes independently by Roichman, and Ahlswede, Bezrukov, Blokhuis, Metsch, and Moorhouse.
APN functions play a big role as primitives in symmetric cryptography as building blocks that yield optimal resistance to differential attacks. In this note, we consider a recent extension, done by Calderini et al. (2023), of a biprojective APN family introduced by Göloğlu (2022) defined on . We show that this generalization yields functions equivalent to Göloğlu's original family if . If we show exactly how many inequivalent APN functions this new family contains. We also show that the family has the minimal image set size for an APN function and determine its Walsh spectrum, hereby settling some open problems. In our proofs, we leverage a group theoretic technique recently developed by Göloğlu and the author in conjunction with a group action on the set of projective polynomials.
For an linear code , the singleton defect of is defined by . When , the code is called a near maximum distance separable (NMDS) code, where is the dual code of . NMDS codes have important applications in finite projective geometries, designs and secret sharing schemes. In this paper, we present four new constructions of infinite families of NMDS codes with dimension 4 and completely determine their weight enumerators. As an application, we also determine the locality of the dual codes of these NMDS codes and obtain four families of distance-optimal and dimension-optimal locally recoverable codes.
In this paper we compute the automorphism group of the curves and introduced in Tafazolian et al. [27] as new examples of maximal curves which cannot be covered by the Hermitian curve. They arise as subcovers of the first generalized GK curve (GGS curve). As a result, a new characterization of the GK curve, as a member of this family, is obtained.
Blokhuis showed that all maximum cliques in Paley graphs of square order have a subfield structure. Recently, it has been shown that in Peisert-type graphs, all maximum cliques are affine subspaces, and yet some maximum cliques do not arise from a subfield. In this paper, we investigate the existence of a clique of size with a subspace structure in pseudo-Paley graphs of order q from unions of semi-primitive cyclotomic classes. We show that such a clique must have an equal contribution from each cyclotomic class and that most such pseudo-Paley graphs do not admit such cliques, suggesting that the Delsarte bound on the clique number can be improved in general. We also prove that generalized Peisert graphs are not isomorphic to Paley graphs or Peisert graphs, confirming a conjecture of Mullin.
It is known that a Bruen chain of the three-dimensional projective space exists for every odd prime power q at most 37, except for . It was shown by Cardinali et al. (2005) that Bruen chains do not exist for . We develop a model, based on finite fields, which allows us to extend this result to , thereby adding more evidence to the conjecture that Bruen chains do not exist for . Furthermore, we show that Bruen chains can be realised precisely as the -cliques of a two related, yet distinct, undirected simple graphs.
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