Large Automorphism Groups Research Articles

A Deterministic Algorithm for Computing the Weight Distribution of Polar Code

In this work, we present a deterministic algorithm for computing the entire weight distribution of polar codes. As the first step, we derive an efficient recursive procedure to compute the weight distribution that arises in successive cancellation decoding of polar codes along any decoding path. This solves the open problem recently posed by Polyanskaya, Davletshin, and Polyanskii. Using this recursive procedure, at code length <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> , we can compute the weight distribution of any <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">polar cosets</i> in time <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ). We show that any polar code can be represented as a disjoint union of such polar cosets; moreover, this representation extends to polar codes with dynamically frozen bits. However, the number of polar cosets in such representation scales exponentially with a parameter introduced herein, which we call the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">mixing factor</i> . To upper bound the complexity of our algorithm for polar codes being decreasing monomial codes, we study the range of their mixing factors. We prove that among all decreasing monomial codes with rates at most 1/2, self-dual Reed-Muller codes have the largest mixing factors. To further reduce the complexity of our algorithm, we make use of the fact that, as decreasing monomial codes, polar codes have a large automorphism group. That automorphism group includes the block lower-triangular affine group (BLTA), which in turn contains the lower-triangular affine group (LTA). We prove that a subgroup of LTA acts transitively on certain subsets of decreasing monomial codes, thereby drastically reducing the number of polar cosets that we need to evaluate. This complexity reduction makes it possible to compute the weight distribution of polar codes at length <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> = 128.

Genuinely nonabelian partial difference sets

AbstractStrongly regular graphs (SRGs) provide a fertile area of exploration in algebraic combinatorics, integrating techniques in graph theory, linear algebra, group theory, finite fields, finite geometry, and number theory. Of particular interest are those SRGs with a large automorphism group. If an automorphism group acts regularly (sharply transitively) on the vertices of the graph, then we may identify the graph with a subset of the group, a partial difference set (PDS), which allows us to apply techniques from group theory to examine the graph. Much of the work over the past four decades has concentrated on abelian PDSs using the powerful techniques of character theory. However, little work has been done on nonabelian PDSs. In this paper we point out the existence of genuinely nonabelian PDSs, that is, PDSs for parameter sets where a nonabelian group is the only possible regular automorphism group. We include methods for demonstrating that abelian PDSs are not possible for a particular set of parameters or for a particular SRG. Four infinite families of genuinely nonabelian PDSs are described, two of which—one arising from triangular graphs and one arising from Krein covers of complete graphs constructed by Godsil—are new. We also include a new nonabelian PDS found by computer search and present some possible future directions of research.

Geometry of smooth extremal surfaces

We study the geometry of smooth projective surfaces defined by Frobenius forms, a class of homogenous polynomials in prime characteristic recently shown to have minimal possible F-pure threshold among forms of the same degree. We call these surfaces extremal surfaces, and show that their geometry is reminiscent of the geometry of smooth cubic surfaces, especially non-Frobenius split cubic surfaces. For instance, extremal surfaces have many lines but no triangles, hence many “star points” analogous to Eckardt points on a cubic surface. We generalize the classical notion of a double six for cubic surfaces to a double 2d on an extremal surface of degree d. We show that, asymptotically in d, smooth extremal surfaces have at least 116d14 double 2d's. A key element of the proofs is the large automorphism group of an extremal surface, which we show to act transitively on many associated sets, such as the set of triples of skew lines on the extremal surface.

Efficient Decoding Algorithm for Binary Quadratic Residue Codes Using Reduced Permutation Sets

The Quadratic Residue (QR) codes have a rich mathematic structure. Unfortunately, their Algebraic Decoding (AD) is not generalizable for all QR codes. In this study, an efficient hard decoding algorithm is proposed to generalize the decoding of the binary systematic Quadratic Residue (QR) codes. The proposed decoder corrects t erroneous bits or less, in the received word, based on a reduced set of permutations derived from the large automorphism group of QR codes. This set of permutations is applied to the received word to move the error positions and trap all of them in redundancy. Then, to evaluate the proposed method, we applied it to many binary QR codes of moderate code length starting with 17 until 113 with reducible and irreducible generator polynomials. The proposed decoder was validated by inserting all possible error patterns, that have t or less erroneous positions, as input of the proposed decoder and the output is always a correct codeword. The complexity study, in terms of the number of operations used, reveals that the light permutation decoding LPD algorithm significantly decreases decoding complexity without performance loss. So, it is qualified to be a good competitor to decode QR codes with lower lengths but is the best for QR codes with higher lengths.

Invariant Measures for Large Automorphism Groups of Projective Surfaces

Invariant Measures for Large Automorphism Groups of Projective Surfaces

On Infinite Families of Narrow-Sense Antiprimitive BCH Codes Admitting 3-Transitive Automorphism Groups and Their Consequences

The Bose-Chaudhuri-Hocquenghem (BCH) codes are a well-studied subclass of cyclic codes that have found numerous applications in error correction and notably in quantum information processing. A subclass of attractive BCH codes is the narrow-sense BCH codes over the Galois field $\mathrm{GF}(q)$ with length $q+1$, which are closely related to the action of the projective general linear group of degree two on the projective line. This paper aims to study some of the codes within this class and specifically narrow-sense antiprimitive BCH codes (these codes are also linear complementary duals (LCD) codes that have interesting practical recent applications in cryptography, among other benefits). We shall use tools and combine arguments from algebraic coding theory, combinatorial designs, and group theory (group actions, representation theory of finite groups, etc.) to investigate narrow-sense antiprimitive BCH Codes and extend results from the recent literature. Notably, the dimension, the minimum distance of some $q$-ary BCH codes with length $q+1$, and their duals are determined in this paper. The dual codes of the narrow-sense antiprimitive BCH codes derived in this paper include almost MDS codes. Furthermore, the classification of $\mathrm{PGL} (2, p^m)$-invariant codes over $\mathrm{GF} (p^h)$ is completed. As an application of this result, the $p$-ranks of all incidence structures invariant under the projective general linear group $\mathrm{ PGL }(2, p^m)$ are determined. Furthermore, infinite families of narrow-sense BCH codes admitting a $3$-transitive automorphism group are obtained. Via these BCH codes, a coding-theory approach to constructing the Witt spherical geometry designs is presented. The BCH codes proposed in this paper are good candidates for permutation decoding, as they have a relatively large group of automorphisms.

On the Efficiency of Polar-Like Decoding for Symmetric Codes

The recently introduced polar codes constitute a breakthrough in coding theory due to their capacity-achieving property. This goes hand in hand with a quasilinear construction, encoding, and successive cancellation list decoding procedures based on the Plotkin construction. The decoding algorithm can be applied with slight modifications to Reed-Muller or eBCH codes, that both achieve the capacity of erasure channels, although the list size needed for good performance grows too fast to make the decoding practical even for moderate block lengths. The key ingredient for proving the capacity-achieving property of Reed-Muller and eBCH codes is their group of symmetries. It can be plugged into the concept of Plotkin decomposition to design various permutation decoding algorithms. Although such techniques allow to outperform the straightforward polar-like decoding, the complexity stays impractical. In this paper, we show that although invariance under a large automorphism group is valuable in a theoretical sense, it also ensures that the list size needed for good performance grows exponentially. We further establish the bounds that arise if we sacrifice some of the symmetries. Although the theoretical analysis of the list decoding algorithm remains an open problem, our result provides an insight into the factors that impact the decoding complexity.

Cartan connections and path structures with large automorphism groups

In this paper, we classify compact manifolds of dimension three equipped with a path structure and a fixed contact form (which we refer to as a strict path structure) under the hypothesis that their automorphism group is non-compact. We use a Cartan connection associated to the structure and show that its curvature is constant.

Gushel–Mukai varieties with many symmetries and an explicit irrational Gushel–Mukai threefold

We construct an explicit smooth Fano complex threefold with Picard number 1, index 1, and degree 10 (also known as a Gushel–Mukai threefold) and prove that it is not rational by showing that its intermediate Jacobian has a faithful \({{\,\mathrm{PSL}\,}}(2,\mathbf{F}_{11}) \)-action. Along the way, we construct Gushel–Mukai varieties of various dimensions with rather large (finite) automorphism groups. The starting point of all these constructions is an Eisenbud–Popescu–Walter sextic with a faithful \({{\,\mathrm{PSL}\,}}(2,\mathbf{F}_{11}) \)-action discovered by the second author in 2013.

A comparison between Suzuki invariant code and one-point Hermitian code

In this paper we compare the performance of two algebraic geometry codes (Suzuki and Hermitian codes) constructed using maximal algebraic curves over [Formula: see text] with large automorphism groups by choosing specific divisors. We discuss their parameters, compare the rate of these codes as well as their relative minimum distances, and we show that both codes are asymptotically good in terms of the rate which is in contrast to their behavior in terms of the relative minimum distance.

Classification of 9-dimensional trilinear alternating forms over GF(2)

Let V be a finite-dimensional vector space over a finite field and let f be a trilinear alternating form over V. For such forms, we introduce two new invariants. Together with a generalized radical polynomial used for classification of forms in dimension 8 over GF(2), they are sufficient to distinguish between all trilinear alternating forms in dimension 9 over GF(2). To prove the completeness of the list of forms, we computed their groups of automorphisms. There are 31 degenerate and 317 nondegenerate forms. We point out some forms with either small or large automorphism group.

Delta-invariants for Fano varieties with large automorphism groups

For a variety [Formula: see text], a big [Formula: see text]-divisor [Formula: see text] and a closed connected subgroup [Formula: see text] we define a [Formula: see text]-invariant version of the [Formula: see text]-threshold. We prove that for a Fano variety [Formula: see text] and a connected subgroup [Formula: see text] this invariant characterizes [Formula: see text]-equivariant uniform [Formula: see text]-stability. We also use this invariant to investigate [Formula: see text]-equivariant [Formula: see text]-stability of some Fano varieties with large groups of symmetries, including spherical Fano varieties. We also consider the case of [Formula: see text] being a finite group.

A new generalization of generalized Petersen graphs

We discuss a new family of cubic graphs, which we call group divisible generalized Petersen graphs (GDGP-graphs), that bears a close resemblance to the family of generalized Petersen graphs; both in definition and properties. The focus of our paper is on determining the algebraic properties of graphs from our new family. We look for highly symmetric graphs, e.g., graphs with large automorphism groups, and vertex- or arc-transitive graphs. In particular, we present arithmetic conditions for the defining parameters that guarantee that graphs with these parameters are vertex-transitive or Cayley, and we find one arc-transitive GDGP-graph which is neither a CQ graph of Feng and Wang, nor a generalized Petersen graph.

Large automorphism groups of ordinary curves of even genus in odd characteristic

Let be a (projective, nonsingular, geometrically irreducible) curve of even genus defined over an algebraically closed field K of odd characteristic p. If the p-rank equals then is ordinary. In this paper, we deal with large automorphism groups G of ordinary curves of even genus. We prove that The proof of our result is based on the classification of automorphism groups of curves of even genus in positive characteristic by Giulietti and Korchmáros. According to this classification, for the exceptional cases and we show that the classical Hurwitz bound holds, unless p = 3, and an example for the latter case being given by the modular curve X(11) in characteristic 3.

On the classification of homogeneous affine tube domains with large automorphism groups in arbitrary dimensions

We classify tube domains in Cn+1 (n≥1) with affinely homogeneous base of their boundary and a.) with positive definite Levi form and b.) with Lorentzian type Levi form and affine isotropy of dimension at least (n−2)(n−3)2.

On the existence of PD-sets: Algorithms arising from automorphism groups of codes

<p style='text-indent:20px;'>This paper deals with the problem of determining whether a PD-set exists for a given linear code <inline-formula><tex-math id="M1">\begin{document}$ \mathcal C $\end{document}</tex-math></inline-formula> and information set <inline-formula><tex-math id="M2">\begin{document}$ I $\end{document}</tex-math></inline-formula>. A computational approach is proposed and illustrated with two exceptional codes with automorphism groups isomorphic to the sporadic simple groups <inline-formula><tex-math id="M3">\begin{document}$ \mathrm{M}_{12} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ \mathrm{M}_{22} $\end{document}</tex-math></inline-formula>, respectively. In both cases, the existence of a PD–set is proven. In general, the algorithm works well whenever the code <inline-formula><tex-math id="M5">\begin{document}$ \mathcal C $\end{document}</tex-math></inline-formula> has a very large automorphism group.

A note on large automorphism groups of compact Riemann surfaces

Belolipetsky and Jones classified those compact Riemann surfaces of genus g admitting a large group of automorphisms of order λ(g−1), for each λ>6, under the assumption that g−1 is a prime number. In this article we study the remaining large cases; namely, we classify Riemann surfaces admitting 5(g−1) and 6(g−1) automorphisms, with g−1 a prime number. As a consequence, we obtain the classification of Riemann surfaces admitting a group of automorphisms of order 3(g−1), with g−1 a prime number. We also provide isogeny decompositions of their Jacobian varieties.

On the Parametrization of an Algebraic Curve

At present, a plane algebraic curve can be parametrized in the following two cases: if its genus is equal to 0 or 1 and if it has a large group of birational automorphisms. Here we propose a new polyhedron method (involving a polyhedron called a Hadamard polyhedron by the author), which allows us to divide the space ℝ2 or ℂ2 into pieces in each of which the polynomial specifying the curve is sufficiently well approximated by its truncated polynomial, which often defines the parametrized curve. This approximate parametrization in a piece can be refined by means of the Newton method. Thus, an arbitrarily exact piecewise parametrization of the original curve can be obtained.

Large automorphism groups of ordinary curves in characteristic 2

Let X be a (projective, non-singular, irreducible) curve of even genus g(X)≥2 defined over an algebraically closed field K of characteristic p. If the p-rank γ(X) equals g(X), then X is ordinary. In this paper, we deal with large automorphism groups G of ordinary curves. Under the hypotheses that p=2, g(X) is even and G is solvable, we prove that |G|<35(g(X)+1)3/2.

Contact metric manifolds with large automorphism group and (κ, µ)-spaces

AbstractWe discuss the classifiation of simply connected, complete (κ, µ)-spaces from the point of view of homogeneous spaces. In particular, we exhibit new models of (κ, µ)-spaces having Boeckx invariant -1. Finally, we prove that the number ${{(n + 1)(n + 2)} \over 2}$ is the maximum dimension of the automorphism group of a contact metric manifold of dimension 2n +1, n ≥ 2, whose symmetric operator h has rank at least 3 at some point; if this dimension is attained, and the dimension of the manifold is not 7, it must be a (κ, µ)-space. The same conclusion holds also in dimension 7 provided the almost CR structure of the contact metric manifold under consideration is integrable.