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 T. 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 quasi-cyclic 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 decibel away from the Shannon theoretical limit with iterative decoding.

We study the problem of learning a binary relation between two sets of objects or between a set and itself. We represent a binary relation between a set of size n and a set of size m as an n m matrix of bits, whose (i � j) entry is 1 if and only if the relation holds between the corresponding elements of the two sets. We present polynomial prediction algorithms for learning binary relations in an extended on-line learning model, where the examples are drawn by the learner, by a helpful teacher, by an adversary, or according to a uniform probability distribution on the instance space. In the rst part of this paper, we present results for the case that the matrix of the relation has at most k row types. We present upper and lower bounds on the number of prediction mistakes any prediction algorithm makes when learning such a matrix under the extended on-line learning model. Furthermore, we describe a technique that simpli es the proof of expected mistake bounds against a randomly chosen query sequence. In the second part of this paper, we consider the problem of learning a binary relation that is a total order on a set. We describe a general technique using a fully

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 Eu-clidean 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 var-ious 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 quasi-cyclic 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.

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

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

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 0-sporadic. Example: Finitely many Chevalley groups are 0-sporadic. A proven result: Among polynomial 0-sporadic groups, precisely three produce covers falling in nontrivial reduced families. Each (miraculously) defines one natural genus 0 Q cover of the j-line. 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 j-line covers in two cases. First: From the three 0-sporadic examples of Problem g=0

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 FpG-module M.

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 bit-length O(log n + log q). Moreover, An and Sn have presentations with 3 generators, 7 relations and bit-length O(log n), while SL(n, q) has a presentation with 6 generators, 25 relations and

Abstract: The schemes for organizing binary-valued records using finite geometries have been extended to the situation in which the attributes of the records can take multiple values. Some new schemes for organizing records have been proposed which are based on deleted finite geometries. These new schemes permit the organization of records into buckets in such a manner that, by solving certain algebraic linear equations over a finite field, it is possible to determine the bucket in which records, pertaining to two given values of two different attributes, are stored. Since the bucket identification required for the storage of record accession numbers is based on the combination of attribute values, the file does not require any reorganization as new records are added. This is a definite advantage of the proposed schemes over many key-address transformation procedures wherein the addition of new records may lead to either a drastic revision of the file organization or significant reduction of retrieval effectiveness. The search time for the new schemes are very small in comparison to other existing methods. 1.

The Matsumoto-Imai 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 Matsumoto-Imai 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