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50
Lowdensity paritycheck codes based on finite geometries: A rediscovery and new results
 IEEE Trans. Inform. Theory
, 2001
"... This paper presents a geometric approach to the construction of lowdensity paritycheck (LDPC) codes. Four classes of LDPC codes are constructed based on the lines and points of Euclidean and projective geometries over finite fields. Codes of these four classes have good minimum distances and thei ..."
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Cited by 139 (6 self)
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This paper presents a geometric approach to the construction of lowdensity paritycheck (LDPC) codes. Four classes of LDPC codes are constructed based on the lines and points of Euclidean and projective geometries over finite fields. Codes of these four classes have good minimum distances and their Tanner graphs have girth T. Finitegeometry LDPC codes can be decoded in various ways, ranging from low to high decoding complexity and from reasonably good to very good performance. They perform very well with iterative decoding. Furthermore, they can be put in either cyclic or quasicyclic form. Consequently, their encoding can be achieved in linear time and implemented with simple feedback shift registers. This advantage is not shared by other LDPC codes in general and is important in practice. Finitegeometry LDPC codes can be extended and shortened in various ways to obtain other good LDPC codes. Several techniques of extension and shortening are presented. Long extended finitegeometry LDPC codes have been constructed and they achieve a performance only a few tenths of a decibel away from the Shannon theoretical limit with iterative decoding.
Low density parity check codes based on finite geometries: A rediscovery and new results
 IEEE Trans. Inform. Theory
, 2001
"... This paper presents a geometric approach to the construction of low density parity check (LDPC) codes. Four classes of LDPC codes are constructed based on the lines and points of Euclidean and projective geometries over finite fields. Codes of these four classes have good minimum distances and thei ..."
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Cited by 37 (11 self)
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This paper presents a geometric approach to the construction of low density parity check (LDPC) codes. Four classes of LDPC codes are constructed based on the lines and points of Euclidean and projective geometries over finite fields. Codes of these four classes have good minimum distances and their Tanner graphs have girth 6. Finite geometry LDPC codes can be decoded in various ways, ranging from low to high decoding complexity and from reasonably good to very good performance. They perform very well with iterative decoding. Furthermore, they can be put in either cyclic or quasicyclic form. Consequently, their encoding can be achieved in linear time and implemented with simple feedback shift registers. This advantage is not shared by other LDPC codes in general and is important in practice. Finite geometry LDPC codes can be extended and shortened in various ways to obtain other good LDPC codes. Several techniques of extension and shortening are presented. Long extended finite geometry LDPC codes have been constructed and they achieve a performance only a few tenths of a dB away from the Shannon theoretical limit with iterative decoding.
Learning binary relations and total orders
 In Proceedings of the 30th Annual IEEE Symposium on Foundations of Computer Science
, 1989
"... Abstract. We study the problem of designing polynomial prediction algorithms for learning binary relations. We study these problems under an online model in which the instances are drawn by the learner, by a helpful teacher, by an adversary or according to a probability distribution on the instance ..."
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Cited by 32 (5 self)
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Abstract. We study the problem of designing polynomial prediction algorithms for learning binary relations. We study these problems under an online model in which the instances are drawn by the learner, by a helpful teacher, by an adversary or according to a probability distribution on the instance space. We represent the relation as an n x m binary matrix, and present results for when the matrix is restricted to have at most k distinct row types, and when it is constrained by requiring that the predicate form a total order. 1
Presentations of finite simple groups: a quantitative approach
"... There is a constant C0 such that all nonabelian finite simple groups of rank n over Fq, with the possible exception of the Ree groups 2G2(32e+1), have presentations with at most C0 generators and relations and total length at most C0(log n + log q). As a corollary, we deduce a conjecture of Holt: th ..."
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Cited by 12 (4 self)
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There is a constant C0 such that all nonabelian finite simple groups of rank n over Fq, with the possible exception of the Ree groups 2G2(32e+1), have presentations with at most C0 generators and relations and total length at most C0(log n + log q). As a corollary, we deduce a conjecture of Holt: there is a constant C such that dim H2 (G, M) ≤ C dim M for every finite simple group G, every prime p and every irreducible FpGmodule M.
Relating two genus 0 problems of John Thompson
 IN PROGRESS IN GALOIS THEORY, H. VOELKLEIN AND T. SHASKA EDITORS 2005 SPRINGER SCIENCE
"... Excluding a precise list of groups like alternating, symmetric, cyclic and dihedral, from 1st year algebra (§7.2.3), we expect there are only finitely many monodromy groups of primitive genus 0 covers. Denote this nearly proven genus 0 problem as Problem g=0 2. We call the exceptional groups 0spo ..."
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Cited by 12 (7 self)
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Excluding a precise list of groups like alternating, symmetric, cyclic and dihedral, from 1st year algebra (§7.2.3), we expect there are only finitely many monodromy groups of primitive genus 0 covers. Denote this nearly proven genus 0 problem as Problem g=0 2. We call the exceptional groups 0sporadic. Example: Finitely many Chevalley groups are 0sporadic. A proven result: Among polynomial 0sporadic groups, precisely three produce covers falling in nontrivial reduced families. Each (miraculously) defines one natural genus 0 Q cover of the jline. The latest Nielsen class techniques apply to these dessins d’enfant to see their subtle arithmetic and interesting cusps. John Thompson earlier considered another genus 0 problem: To find θfunctions uniformizing certain genus 0 (near) modular curves. We call this Problem g=0 1. We pose uniformization problems for jline covers in two cases. First: From the three 0sporadic examples of Problem g=0
Rigidity and real residue class fields
 Acta Arith
, 1990
"... Introduction and acknowledgements: Consider a cover ϕ: X →P 1 x of the Riemann sphere (uniformized by x) by a projective nonsingular curve X with r>2 branch points. Assume that both the curves and the map are defined over Q. Generalizing Serre [Se] we consider not necessarily Galois covers with a ..."
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Cited by 11 (6 self)
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Introduction and acknowledgements: Consider a cover ϕ: X →P 1 x of the Riemann sphere (uniformized by x) by a projective nonsingular curve X with r>2 branch points. Assume that both the curves and the map are defined over Q. Generalizing Serre [Se] we consider not necessarily Galois covers with any number r of branch points (not necessarily in R). We show how to compute the action of complex conjugation on the fiber in X over a real value of x0 ∈P 1 x. It is an “exceptional cover ” for which all of the residue class
On three Engel groups
 Bull. Austral. Math. Soc
, 1972
"... Dedicated to Hermann Heineken on the occasion of his 70th birthday Commutators originated over 100 years ago as a byproduct of computing group characters of nonabelian groups. They are now an established and immensely useful tool in all of group theory. Commutators became objects of interest in the ..."
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Cited by 11 (0 self)
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Dedicated to Hermann Heineken on the occasion of his 70th birthday Commutators originated over 100 years ago as a byproduct of computing group characters of nonabelian groups. They are now an established and immensely useful tool in all of group theory. Commutators became objects of interest in their own right soon after their introduction. In particular, the phenomenon that the set of commutators does not necessarily form a subgroup has been well documented with various kinds of examples. Many of the early results have been forgotten and were rediscovered over the years. In this paper we give a historical overview of the origins of commutators and a survey of different kinds of groups where the set of commutators does not equal the commutator subgroup. We conclude with a status report on what is now called the Ore Conjecture stating that every element in a finite nonabelian simple group is a commutator. 1 Origins of commutators “In a group the product of two commutators need not be a commu
Presentations of finite simple groups: a computational approach
"... All nonabelian finite simple groups of Lie type of rank n over a field of size q, with the possible exception of the Ree groups 2 G2(q), have presentations with at most 49 relations and bitlength O(log n + log q). Moreover, An and Sn have presentations with 3 generators, 7 relations and bitlength ..."
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Cited by 6 (4 self)
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All nonabelian finite simple groups of Lie type of rank n over a field of size q, with the possible exception of the Ree groups 2 G2(q), have presentations with at most 49 relations and bitlength O(log n + log q). Moreover, An and Sn have presentations with 3 generators, 7 relations and bitlength O(log n), while SL(n, q) has a presentation with 6 generators, 25 relations and
Obtaining the neutrino mixing matrix with the tetrahedral group, Phys
 Lett. B630
"... We discuss various “minimalist ” schemes to derive the neutrino mixing matrix using the tetrahedral group A4. ..."
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Cited by 5 (0 self)
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We discuss various “minimalist ” schemes to derive the neutrino mixing matrix using the tetrahedral group A4.