4,082 research outputs found

    Applications of finite geometries to designs and codes

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    This dissertation concerns the intersection of three areas of discrete mathematics: finite geometries, design theory, and coding theory. The central theme is the power of finite geometry designs, which are constructed from the points and t-dimensional subspaces of a projective or affine geometry. We use these designs to construct and analyze combinatorial objects which inherit their best properties from these geometric structures. A central question in the study of finite geometry designs is Hamada’s conjecture, which proposes that finite geometry designs are the unique designs with minimum p-rank among all designs with the same parameters. In this dissertation, we will examine several questions related to Hamada’s conjecture, including the existence of counterexamples. We will also study the applicability of certain decoding methods to known counterexamples. We begin by constructing an infinite family of counterexamples to Hamada’s conjecture. These designs are the first infinite class of counterexamples for the affine case of Hamada’s conjecture. We further demonstrate how these designs, along with the projective polarity designs of Jungnickel and Tonchev, admit majority-logic decoding schemes. The codes obtained from these polarity designs attain error-correcting performance which is, in certain cases, equal to that of the finite geometry designs from which they are derived. This further demonstrates the highly geometric structure maintained by these designs. Finite geometries also help us construct several types of quantum error-correcting codes. We use relatives of finite geometry designs to construct infinite families of q-ary quantum stabilizer codes. We also construct entanglement-assisted quantum error-correcting codes (EAQECCs) which admit a particularly efficient and effective error-correcting scheme, while also providing the first general method for constructing these quantum codes with known parameters and desirable properties. Finite geometry designs are used to give exceptional examples of these codes

    Hadamard 2-(63,31,15) designs invariant under the dihedral group of order 10

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    AbstractAll Hadamard 2-(63,31,15) designs invariant under the dihedral group of order 10 are constructed and classified up to isomorphism together with related Hadamard matrices of order 64. Affine 2-(64,16,5) designs can be obtained from Hadamard 2-(63,31,15) designs having line spreads by Rahilly’s construction [A. Rahilly, On the line structure of designs, Discrete Math. 92 (1991) 291–303]. The parameter set 2-(64,16,5) is one of two known sets when there exists several nonisomorphic designs with the same parameters and p-rank as the design obtained from the points and subspaces of a given dimension in affine geometry AG(n,pm) (p a prime). It is established that an affine 2-(64,16,5) design of 2-rank 16 that is associated with a Hadamard 2-(63,31,15) design invariant under the dihedral group of order 10 is either isomorphic to the classical design of the points and hyperplanes in AG(3,4), or is one of the two exceptional designs found by Harada, Lam and Tonchev [M. Harada, C. Lam, V.D. Tonchev, Symmetric (4, 4)-nets and generalized Hadamard matrices over groups of order 4, Designs Codes Cryptogr. 34 (2005) 71–87]

    Entanglement-assisted quantum low-density parity-check codes

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    This paper develops a general method for constructing entanglement-assisted quantum low-density parity-check (LDPC) codes, which is based on combinatorial design theory. Explicit constructions are given for entanglement-assisted quantum error-correcting codes (EAQECCs) with many desirable properties. These properties include the requirement of only one initial entanglement bit, high error correction performance, high rates, and low decoding complexity. The proposed method produces infinitely many new codes with a wide variety of parameters and entanglement requirements. Our framework encompasses various codes including the previously known entanglement-assisted quantum LDPC codes having the best error correction performance and many new codes with better block error rates in simulations over the depolarizing channel. We also determine important parameters of several well-known classes of quantum and classical LDPC codes for previously unsettled cases.Comment: 20 pages, 5 figures. Final version appearing in Physical Review

    Coding Theory and Algebraic Combinatorics

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    This chapter introduces and elaborates on the fruitful interplay of coding theory and algebraic combinatorics, with most of the focus on the interaction of codes with combinatorial designs, finite geometries, simple groups, sphere packings, kissing numbers, lattices, and association schemes. In particular, special interest is devoted to the relationship between codes and combinatorial designs. We describe and recapitulate important results in the development of the state of the art. In addition, we give illustrative examples and constructions, and highlight recent advances. Finally, we provide a collection of significant open problems and challenges concerning future research.Comment: 33 pages; handbook chapter, to appear in: "Selected Topics in Information and Coding Theory", ed. by I. Woungang et al., World Scientific, Singapore, 201

    Repairable Replication-based Storage Systems Using Resolvable Designs

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    We consider the design of regenerating codes for distributed storage systems at the minimum bandwidth regeneration (MBR) point. The codes allow for a repair process that is exact and uncoded, but table-based. These codes were introduced in prior work and consist of an outer MDS code followed by an inner fractional repetition (FR) code where copies of the coded symbols are placed on the storage nodes. The main challenge in this domain is the design of the inner FR code. In our work, we consider generalizations of FR codes, by establishing their connection with a family of combinatorial structures known as resolvable designs. Our constructions based on affine geometries, Hadamard designs and mutually orthogonal Latin squares allow the design of systems where a new node can be exactly regenerated by downloading β≥1\beta \geq 1 packets from a subset of the surviving nodes (prior work only considered the case of β=1\beta = 1). Our techniques allow the design of systems over a large range of parameters. Specifically, the repetition degree of a symbol, which dictates the resilience of the system can be varied over a large range in a simple manner. Moreover, the actual table needed for the repair can also be implemented in a rather straightforward way. Furthermore, we answer an open question posed in prior work by demonstrating the existence of codes with parameters that are not covered by Steiner systems

    Counting Steiner triple systems with classical parameters and prescribed rank

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    By a famous result of Doyen, Hubaut and Vandensavel \cite{DHV}, the 2-rank of a Steiner triple system on 2n−12^n-1 points is at least 2n−1−n2^n -1 -n, and equality holds only for the classical point-line design in the projective geometry PG(n−1,2)PG(n-1,2). It follows from results of Assmus \cite{A} that, given any integer tt with 1≤t≤n−11 \leq t \leq n-1, there is a code Cn,tC_{n,t} containing representatives of all isomorphism classes of STS(2n−1)(2^n-1) with 2-rank at most 2n−1−n+t2^n -1 -n + t. Using a mixture of coding theoretic, geometric, design theoretic and combinatorial arguments, we prove a general formula for the number of distinct STS(2n−1)(2^n-1) with 2-rank at most 2n−1−n+t2^n -1 -n + t contained in this code. This generalizes the only previously known cases, t=1t=1, proved by Tonchev \cite{T01} in 2001, t=2t=2, proved by V. Zinoviev and D. Zinoviev \cite{ZZ12} in 2012, and t=3t=3 (V. Zinoviev and D. Zinoviev \cite{ZZ13}, \cite{ZZ13a} (2013), D. Zinoviev \cite{Z16} (2016)), while also unifying and simplifying the proofs. This enumeration result allows us to prove lower and upper bounds for the number of isomorphism classes of STS(2n−1)(2^n-1) with 2-rank exactly (or at most) 2n−1−n+t2^n -1 -n + t. Finally, using our recent systematic study of the ternary block codes of Steiner triple systems \cite{JT}, we obtain analogous results for the ternary case, that is, for STS(3n)(3^n) with 3-rank at most (or exactly) 3n−1−n+t3^n -1 -n + t. We note that this work provides the first two infinite families of 2-designs for which one has non-trivial lower and upper bounds for the number of non-isomorphic examples with a prescribed pp-rank in almost the entire range of possible ranks.Comment: 27 page
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