949 research outputs found
Symplectic spreads, planar functions and mutually unbiased bases
In this paper we give explicit descriptions of complete sets of mutually
unbiased bases (MUBs) and orthogonal decompositions of special Lie algebras
obtained from commutative and symplectic semifields, and
from some other non-semifield symplectic spreads. Relations between various
constructions are also studied. We show that the automorphism group of a
complete set of MUBs is isomorphic to the automorphism group of the
corresponding orthogonal decomposition of the Lie algebra .
In the case of symplectic spreads this automorphism group is determined by the
automorphism group of the spread. By using the new notion of pseudo-planar
functions over fields of characteristic two we give new explicit constructions
of complete sets of MUBs.Comment: 20 page
Euclidean weights of codes from elliptic curves over rings
We construct certain error-correcting codes over finite rings and estimate their parameters. For this purpose, we need to develop some tools, notably an estimate for certain exponential sums and some results on canonical lifts of elliptic curves. These results may be of independent interest.
A code is a subset of An, where A is a finite set (called the alphabet). Usually A is just the field of two elements and, in this case, one speaks of binary codes. Such codes are used in applications where one transmits information through noisy channels. By building redundancy into the code, transmitted messages can be recovered at the receiving end. A code has parameters that measure its eficiency and error-correcting capability. For various reasons one often restricts attention to linear codes, which are linear subspaces of An when A is a field. However, there are non-linear binary codes (such as the Nordstrom-Robinson, Kerdock, and Preparata codes) that outperform linear codes for certain parameters. These codes have remained somewhat mysterious until recently when Hammons, et al. ([6]) discovered that one can obtain these codes from linear codes over rings (i.e. submodules of An, A a ring) via the Gray mapping, which we recall below. In a difierent vein, over the last decade there has been a lot of interest in linear codes coming from algebraic curves over finite fields. The construction of such codes was first proposed by Goppa in [5]; see [15] or [16] for instance. In [17], it is proven that for q ≥ 49 a square, there exist sequences of codes over the finite field with q elements which give asymptotically the best known linear codes over these fields. The second author has extended Goppa\u27s construction to curves over local Artinian rings and shown, for instance, that the Nordstrom-Robinson code can be obtained from her construction followed by the Gray mapping; see [20] and [21]. While most of the parameters for these new codes were estimated in the above papers, the crucial parameter needed to describe the error-correcting capability of the images of these codes under the Gray mapping was still lacking. In this paper we consider the second author\u27s construction in the special case of elliptic curves which are defined over finite local rings and which are the canonical lifts of their reductions. (See section 4 for more about canonical lifts.) For these codes, the missing parameter can be estimated, and we do so
Special subvarieties of Drinfeld modular varieties
We explore an analogue of the Andr\'e-Oort conjecture for subvarieties of
Drinfeld modular varieties. The conjecture states that a subvariety of a
Drinfeld modular variety contains a Zariski-dense set of complex multiplication
(CM) points if and only if is a "special" subvariety (i.e. is defined
by requiring additional endomorphisms). We prove this conjecture in two cases.
Firstly when contains a Zariski-dense set of CM points with a certain
behaviour above a fixed prime (which is the case if these CM points lie in one
Hecke orbit), and secondly when is a curve containing infinitely many CM
points without any additional assumptions.Comment: 22 pages, significant rewrit
Security considerations for Galois non-dual RLWE families
We explore further the hardness of the non-dual discrete variant of the
Ring-LWE problem for various number rings, give improved attacks for certain
rings satisfying some additional assumptions, construct a new family of
vulnerable Galois number fields, and apply some number theoretic results on
Gauss sums to deduce the likely failure of these attacks for 2-power cyclotomic
rings and unramified moduli
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