Topic
Ring of integers
About: Ring of integers is a research topic. Over the lifetime, 1856 publications have been published within this topic receiving 15882 citations.
Papers published on a yearly basis
Papers
More filters
TL;DR: Given an integer m, a product of distinct primes p i, a method is given for constructing codes over the ring of integers modulo m from cyclic codes over GF ( p i ).
Abstract: Given an integer m which is a product of distinct primes p i , a method is given for constructing codes over the ring of integers modulo m from cyclic codes over GF ( p i ). Specifically, if we are given a cyclic ( n , k i ) code over GF ( p i ) with minimum Hamming distance d i , for each i , then we construct a code of block length n over the integers modulo m with π i p k i i codewords, which is both linear and cyclic and has minimum Hamming distance min i d i .
124 citations
TL;DR: In this paper, the construction of BCH codes over finite fields was shown to be similar to that for BCH code over finite integer rings, where the generator polynomials are derived by factorization of x n −1 over the unit ring of an appropriate extension of the Finite integer ring.
Abstract: Bose-Chadhuri-Hocquenghem (BCH) codes with symbols from an arbitrary finite integer ring are derived in terms of their generator polynomials. Tile derivation is based on the factorization of x^{n}-1 over the unit ring of an appropriate extension of the Finite integer ring. The construction is thus shown to be similar to that for BCH codes over finite fields.
122 citations
TL;DR: In this article, the relation between the Jacobian of a projective curve and the type of its Jacobian has been studied in the context of the Picard functor Pic X/S. The relation between Jacobian and projective projective curves has been shown to be equivalent to the equivalence between the Artin-Tate and Birch-SwinnertonDyer conjectures.
Abstract: Let K be a discrete valuation field. Let OK denote the ring of integers of K , and let k be the residue field of OK , of characteristic p ≥ 0. Let S := SpecOK . Let X K be a smooth geometrically connected projective curve of genus 1 over K . Denote by EK the Jacobian of X K . Let X/S and E/S be the minimal regular models of X K and EK , respectively. In this article, we investigate the possible relationships between the special fibers Xk and Ek . In doing so, we are led to study the geometry of the Picard functor Pic X/S when X/S is not necessarily cohomologically flat. As an application of this study, we are able to prove in full generality a theorem of Gordon on the equivalence between the Artin-Tate and Birch-SwinnertonDyer conjectures. Recall that when k is algebraically closed, the special fibers of elliptic curves are classified according to their Kodaira type, which is denoted by a symbol T ∈ {In, I∗n, n ∈ Z≥0, II, II∗, III, III∗, IV, IV∗}. Given a type T and a positive integer m, we denote by mT the new type obtained from T by multiplying all the multiplicities of T by m. When k is algebraically closed, the relationships between the type of a curve of genus 1 and the type of its Jacobian can be summarized as follows.
118 citations
TL;DR: A systematic numerical search for genus two curves defined over the rationals such that their Jacobians are simple and have endomorphism ring equal to the ring of integers of a quartic CM field finds 19 non-isomorphic such curves.
Abstract: We present the results of a systematic numerical search for genus two curves defined over the rationals such that their Jacobians are simple and have endomorphism ring equal to the ring of integers of a quartic CM field. Including the well-known example y 2 = x 5 - 1 we find 19 non-isomorphic such curves. We believe that these are the only such curves.
115 citations
TL;DR: Linear codes over the ring of integers modulo q = pr, p a prime, are considered and natural analogs to Hamming, Reed—Solomon, and BCH codes over finite fields are defined and their properties investigated.
Abstract: Linear codes over the ring of integers modulo q = pr, p a prime, are considered. Natural analogs to Hamming, Reed—Solomon, and BCH codes over finite fields are defined and their properties investigated. Some ring theoretic problems encountered are discussed.
114 citations