1,721,011 research outputs found
Connections between scattered linear sets and MRD-codes
No full textThe aim of this paper is to survey on the known results on maximum scattered linear sets and MRD-codes. In particular, we investigate the link between these two areas. In [57] Sheekey showed how maximum scattered linear sets of PG(1,qn) define square MRD-codes. Later in [13] maximum scattered linear sets in PG(r - 1, qn), r = 2, were used to construct non square MRD-codes. Here, we point out a new relation regarding the other direction. We also provide an alternative proof of the well-known Blokhuis-Lavrauw’s bound for the rank of maximum scattered linear sets shown in [6]
On the number of roots of some linearized polynomials
No full textLinearized polynomials appear in many different contexts, such as rank metric codes, cryptography and linear sets, and the main issue regards the characterization of the number of roots from their coefficients. Results of this type have been already proved in [7,10,24]. In this paper we provide bounds and characterizations on the number of roots of linearized polynomials of this form ax+b0xqjavax.xml.bind.JAXBElement@7412a953+b1xqjavax.xml.bind.JAXBElement@752ac3ad+b2xqjavax.xml.bind.JAXBElement@4a94cef3+...+bt−1xqjavax.xml.bind.JAXBElement@6f9ed9c5∈Fqjavax.xml.bind.JAXBElement@25eb4dc3[x], with gcd(s,n)=1. Also, we characterize the number of roots of such polynomials directly from their coefficients, dealing with matrices which are much smaller than the relative Dickson matrices and the companion matrices used in the previous papers. Furthermore, we develop a method to find explicitly the roots of a such polynomial by finding the roots of a qn-polynomial. Finally, as an application of the above results, we present a family of linear sets of the projective line whose points have a small spectrum of possible weights, containing most of the known families of scattered linear sets. In particular, we carefully study the linear sets in PG(1,q6) presented in [9]
Full weight spectrum one-orbit cyclic subspace codes
No full textFor a linear Hamming metric code of length n over a finite field, the number of distinct weights of its codewords is at most n. The codes achieving the equality in the above bound were called full weight spectrum codes. In this paper, we will focus on the analogous class of codes within the framework of cyclic subspace codes. Cyclic subspace codes have garnered significant attention, particularly for their applications in random network coding to correct errors and erasures. We investigate one-orbit cyclic subspace codes that are full weight spectrum in this context. Utilizing number- theoretical results and combinatorial arguments, we provide a complete classification of full weight spectrum one-orbit cyclic subspace codes. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/)
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