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Mordell-Weil groups and automorphism groups of elliptic K 3 surfaces
We present a method to calculate the action of the Mordell--Weil group of an elliptic K3 surface on the numerical Neroti--Severi lattice of the K3 surface. As an application, we compute a finite generating set of the automorphism group of a K3 surface birational to the double plane branched along a 6-cuspidal sextic curve of torus type. Keywords: K3 surface, double plane, automorphism group, Mordell--Weil group, hyperbolic lattice.
Journal Article
Code generation with Roslyn
\"Learn how Roslyn's new code generation capability will let you write software that is more concise, runs faster, and is easier to maintain. You will learn from real-world business applications to create better software by letting the computer write its own code based on your business logic already defined in lookup tables. Code Generation with Rosyln is the first book to cover this new capability. You will learn how these techniques can be used to simplify systems integration so that if one system already defines business logic through lookup tables, you can integrate a new system and share business logic by allowing the new system to write its own business logic based on already existing table-based business logic. One of the many benefits you will discover is that Roslyn uses an innovative approach to compiler design, opening up the inner workings of the compiler process. You will learn how to see the syntax tree that Roslyn is building as it compiles your code. Additionally, you will learn to feed it your own syntax tree that you create on the fly.\"-- Provided by publisher
Book
On Optimal and Quantum Code Construction from Cyclic Codes over F sub.qIPQ/I with Applications
The key objective of this paper is to study the cyclic codes over mixed alphabets on the structure of F[sub.q]PQ, where P=Fq[v]/〈v3−α22v〉 and Q=Fq[u,v]/〈u2−α12,v3−α22v〉 are nonchain finite rings and α[sub.i] is in F[sub.q]/0 for i∈1,2, where q=p[sup.m] with m≥1 is a positive integer and p is an odd prime. Moreover, with the applications, we obtain better and new quantum error-correcting (QEC) codes. For another application over the ring P, we obtain several optimal codes with the help of the Gray image of cyclic codes.
Journal Article
Structures, Ranks and Minimal Distances of Cyclic Codes over Zsub.p2+uZsub.p2
Let p be a prime and F[sub.p] a finite field of order p. This paper investigates cyclic codes over the ring R[sub.p2,u]=Z[sub.p2]+uZ[sub.p2] of order p[sup.4], where the nilpotent element u satisfies u[sup.2]=0 and pu≠0. The condition u[sup.2]=0 with pu≠0 is crucial, as it creates a nontrivial interaction between the components of the ring, allowing the construction of new codes with enhanced structural and distance properties. We provide explicit generating sets for cyclic codes over R[sub.p2,u] and study fundamental parameters such as their rank and Hamming distance. In the case gcd(n,p)=1, we show that cyclic codes can be generated by just two polynomials, which allows a complete determination of their rank and minimal Hamming distance distributions. Furthermore, using the Gray map from R[sub.p2,u] to F[sub.p] [sup.4], we construct all but one of the ternary optimal codes of length 12 as images of cyclic codes over R[sub.32,u], with computations verified using the Magma system.
Journal Article
IRREDUCIBLE GENERATING SETS OF COMPLETE SEMIGROUPS OF UNIONS B.sub.X
In complete semigroups of unions [B.sub.X](D) defined by semilattices of the class [[EPSILON].sub.1](X,4), we describe the set of all external elements and show that it is a generating (and, therefore, irreducible) set of the semigroup [B.sub.X](D). For a finite semigroup [B.sub.X](D), we given a formula for calculating the number of elements of the generating set.
Journal Article
GENERATING SETS OF THE COMPLETE SEMIGROUP OF BINARY RELATIONS DEFINED BY SEMILATTICES OF THE CLASS SIGMA.sub.1
In this paper, we study generating sets of the complete semigroup of binary relations defined by X\"=semilattices of unions of the class [[SIGMA].sub.1](X, 6). Keywords and phrases: semilattice of unions, complete semigroup of binary relations, generating set, quasinormal representation of binary relations. AMS Subject Classification: 20M05, 20M10
Journal Article
GENERATING SETS OF THE COMPLETE SEMIGROUP OF BINARY RELATIONS DEFINED BY SEMILATTICES OF THE CLASS sigma.sub.4
In this paper, we study generating sets of complete semigroups of binary relations defined by X-semilattices of unions of the class [[sigma].sub.4] (X, 4). We find irreducible generating sets for given semigroups.
Journal Article