Advances in Mathematics of Communications doi:10.3934/amc.2022096 ON CYCLIC AND NEGACYCLIC CODES WITH ONE-DIMENSIONAL HULLS AND THEIR APPLICATIONS Steven T. Dougherty ∗1 and Serap S ¸ahinkaya 1,2 1 Department of Mathematics, University of Scranton, Scranton, PA, 18510, USA 2 Tarsus University, Faculty of Engineering Department of Natural and Mathematical Sciences, Mersin, Turkey (Communicated by Sihem Mesnager) Abstract. Linear codes over finite fields with small dimensional hulls have received much attention due to their applications in cryptology and quantum computing. In this paper, we study cyclic and negacyclic codes with one- dimensional hulls. We determine precisely when cyclic and negacyclic codes over finite fields with one-dimensional hulls exist. We also introduce one- dimensional linear complementary pairs of cyclic and negacyclic codes. As an application, we obtain numerous optimal or near optimal cyclic codes with one- dimensional hulls over different fields and, by using these codes, we present new entanglement-assisted quantum error-correcting codes (EAQECCs). In partic- ular, some of these EAQEC codes are maximal distance separable (MDS). We also obtain one-dimensional linear complementary pairs of cyclic codes, which are either optimal or near optimal. 1. Introduction. The notion of the hull of a code was first defined by Assmus and Key in 1990 in order to study the codes generated by the characteristic functions of the lines of a finite projective plane. The hull of a linear code C , denoted by Hull(C ), is its intersection with its dual code, that is Hull(C )= C∩C ⊥ , where C ⊥ is the Euclidean dual of the code C . This notion is an important tool for determining the automorphism group of a linear code and testing the complexity of some algorithms for checking permutation equivalence of two linear codes [24, 25, 38, 39]. More precisely, the algorithms that check the permutation equivalence work well if the size of the hull is small. Many papers have been devoted to linear codes with small hulls such as LCD codes, and linear codes with one-dimensional hulls due to their importance in applications [12, 17, 26, 32, 33, 37, 34]. Another motivation for studying the hulls of linear codes comes from entanglement-assisted quantum error-correcting codes (EAQECCs). In [5], Brun et al. introduced entanglement-assisted quantum error-correcting codes (EAQECCs), which allow the use of classical error-correcting codes without self-orthogonality. They proved that if shared entanglement is available between the sender and re- ceiver in advance, non-self-orthogonal classical codes can be used to construct EAQECCs. Many researchers presented constructions of good EAQECCs, see for example [3, 18, 20, 21, 29, 31, 36, 41, 42]. An [[n, k, d; c]] q EAQECC encodes k 2020 Mathematics Subject Classification. Primary: 11T71, 94B15. Key words and phrases. Cyclic code, nega-cyclic code, hull. * Corresponding author: Steven T. Dougherty. 1