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39
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 119 (4 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 35 (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 34 (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
Hermitian varieties in a finite projective space P~N,q2
 Canadian J. Math
"... The elements x and x q of G.F(q2) are defined to be conjugate to one another. The square Il8trix H = «h ij » is called Hermitian if h ij is conjugate to h ji for all i, j. If!.T:: (xo,~,..., ~T) ' then!.(q) is q q q defined to be the colwnn vector whose elements are x o ':Ki ' •• • , XJ.Ii. A Hermit ..."
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Cited by 13 (2 self)
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The elements x and x q of G.F(q2) are defined to be conjugate to one another. The square Il8trix H = «h ij » is called Hermitian if h ij is conjugate to h ji for all i, j. If!.T:: (xo,~,..., ~T) ' then!.(q) is q q q defined to be the colwnn vector whose elements are x o ':Ki ' •• • , XJ.Ii. A Hermitian variety V N _ l in the finite projective space PG(N, q2) has the equation!.T H x(q) = 0, where H is a Hermitian matrix of order N+l. The present paper studies the geometrical properties of Hermitian varieties. The theory of pole andpolars has been developed, and the sections of these I I
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 (5 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 any ..."
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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
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 7 (5 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.
Fast cryptanalysis of the MatsumotoImai public key scheme
 Advances in Cryptology – EuroCrypt’84, volume 209 of Lecture Notes in Computer Science
, 1985
"... The MatsumotoImai public key scheme was developed to provide very fast signatures. It is based on substitution polynomials over GF ( 2 m). This paper shows in two ways that the MatsumotoImai public key scheme is very easy to break. In the faster of the two attacks the time to cryptanalyze the sche ..."
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Cited by 4 (0 self)
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The MatsumotoImai public key scheme was developed to provide very fast signatures. It is based on substitution polynomials over GF ( 2 m). This paper shows in two ways that the MatsumotoImai public key scheme is very easy to break. In the faster of the two attacks the time to cryptanalyze the scheme is about proportional to the binary length of the public key. This shows that Matsumoto and Imai greatly