Book•
Distance-Regular Graphs
23 Jun 1989-
TL;DR: In this paper, a connected simple graph with vertex set X of diameter d is considered, and the authors define Ri X2 by (x, y) Ri whenever x and y have graph distance.
Abstract: Consider a connected simple graph with vertex set X of diameter d. Define Ri X2 by (x, y) Ri whenever x and y have graph distance
Citations
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TL;DR: Certain notorious nonlinear binary codes contain more codewords than any known linear code and can be very simply constructed as binary images under the Gray map of linear codes over Z/sub 4/, the integers mod 4 (although this requires a slight modification of the Preparata and Goethals codes).
Abstract: Certain notorious nonlinear binary codes contain more codewords than any known linear code. These include the codes constructed by Nordstrom-Robinson (1967), Kerdock (1972), Preparata (1968), Goethals (1974), and Delsarte-Goethals (1975). It is shown here that all these codes can be very simply constructed as binary images under the Gray map of linear codes over Z/sub 4/, the integers mod 4 (although this requires a slight modification of the Preparata and Goethals codes). The construction implies that all these binary codes are distance invariant. Duality in the Z/sub 4/ domain implies that the binary images have dual weight distributions. The Kerdock and "Preparata" codes are duals over Z/sub 4/-and the Nordstrom-Robinson code is self-dual-which explains why their weight distributions are dual to each other. The Kerdock and "Preparata" codes are Z/sub 4/-analogues of first-order Reed-Muller and extended Hamming codes, respectively. All these codes are extended cyclic codes over Z/sub 4/, which greatly simplifies encoding and decoding. An algebraic hard-decision decoding algorithm is given for the "Preparata" code and a Hadamard-transform soft-decision decoding algorithm for the I(Kerdock code. Binary first- and second-order Reed-Muller codes are also linear over Z/sub 4/, but extended Hamming codes of length n/spl ges/32 and the Golay code are not. Using Z/sub 4/-linearity, a new family of distance regular graphs are constructed on the cosets of the "Preparata" code. >
1,347 citations
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TL;DR: In this paper, it was shown that all the nonlinear binary codes constructed by Nordstrom-Robinson, Kerdock, Preparata, Goethals, and Delsarte-Goethals can be constructed as binary images under the Gray map of linear codes over Z_4, the integers mod 4.
Abstract: Certain notorious nonlinear binary codes contain more codewords than any known linear code. These include the codes constructed by Nordstrom-Robinson, Kerdock, Preparata, Goethals, and Delsarte-Goethals. It is shown here that all these codes can be very simply constructed as binary images under the Gray map of linear codes over Z_4, the integers mod 4 (although this requires a slight modification of the Preparata and Goethals codes). The construction implies that all these binary codes are distance invariant. Duality in the Z_4 domain implies that the binary images have dual weight distributions. The Kerdock and "Preparata" codes are duals over Z_4 -- and the Nordstrom-Robinson code is self-dual -- which explains why their weight distributions are dual to each other. The Kerdock and "Preparata" codes are Z_4-analogues of first-order Reed-Muller and extended Hamming codes, respectively. All these codes are extended cyclic codes over Z_4, which greatly simplifies encoding and decoding. An algebraic hard-decision decoding algorithm is given for the "Preparata" code and a Hadamard-transform soft-decision decoding algorithm for the Kerdock code. Binary first- and second-order Reed-Muller codes are also linear over Z_4, but extended Hamming codes of length n >= 32 and the Golay code are not. Using Z_4-linearity, a new family of distance regular graphs are constructed on the cosets of the "Preparata" code.
1,134 citations
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TL;DR: A Reed-Solomon-like code construction, related to Gabidulin's construction of maximum rank-distance codes, is described and a Sudan-style ldquolist-1rdquo minimum-distance decoding algorithm is provided.
Abstract: The problem of error-control in random linear network coding is considered. A ldquononcoherentrdquo or ldquochannel obliviousrdquo model is assumed where neither transmitter nor receiver is assumed to have knowledge of the channel transfer characteristic. Motivated by the property that linear network coding is vector-space preserving, information transmission is modeled as the injection into the network of a basis for a vector space V and the collection by the receiver of a basis for a vector space U. A metric on the projective geometry associated with the packet space is introduced, and it is shown that a minimum-distance decoder for this metric achieves correct decoding if the dimension of the space V capU is sufficiently large. If the dimension of each codeword is restricted to a fixed integer, the code forms a subset of a finite-field Grassmannian, or, equivalently, a subset of the vertices of the corresponding Grassmann graph. Sphere-packing and sphere-covering bounds as well as a generalization of the singleton bound are provided for such codes. Finally, a Reed-Solomon-like code construction, related to Gabidulin's construction of maximum rank-distance codes, is described and a Sudan-style ldquolist-1rdquo minimum-distance decoding algorithm is provided.
1,121 citations
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TL;DR: The application of Grassmannian frames to wireless communication and to multiple description coding is discussed and their connection to unit norm tight frames for frames which are generated by group-like unitary systems is discussed.
1,051 citations
01 Jan 1996
TL;DR: It is showed that many iterative decoding algorithms are special cases of two generic algorithms, the min-sum and sum-product algorithms, which also include non-iterative algorithms such as Viterbi decoding.
Abstract: Iterative decoding techniques have become a viable alternative for constructing high performance coding systems. In particular, the recent success of turbo codes indicates that performance close to the Shannon limit may be achieved. In this thesis, it is showed that many iterative decoding algorithms are special cases of two generic algorithms, the min-sum and sum-product algorithms, which also include non-iterative algorithms such as Viterbi decoding. The min-sum and sum-product algorithms are developed and presented as generalized trellis algorithms, where the time axis of the trellis is replaced by an arbitrary graph, the “Tanner graph”. With cycle-free Tanner graphs, the resulting decoding algorithms (e.g., Viterbi decoding) are maximum-likelihood but suffer from an exponentially increasing complexity. Iterative decoding occurs when the Tanner graph has cycles (e.g., turbo codes); the resulting algorithms are in general suboptimal, but significant complexity reductions are possible compared to the cycle-free case. Several performance estimates for iterative decoding are developed, including a generalization of the union bound used with Viterbi decoding and a characterization of errors that are uncorrectable after infinitely many decoding iterations.
1,044 citations
References
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01 Jan 1966TL;DR: In this paper, the authors highlight equilateral point sets in elliptic geometry and show that Paley's construction may be reversed to obtain a C -matrix of order 46, in view of the existence of a Hadamard matrix of order 92.
Abstract: This chapter highlights equilateral point sets in elliptic geometry. Elliptic space of r−1 dimensions E r−1 is obtained from r -dimensional vector space R r with inner product ( a , b ). For 1 , any k -dimensional linear subspace R k of R r is called a ( k−1 )-dimensional elliptic subspace E k−1 . The query for equilateral point sets in elliptic geometry leads to the search for matrices B of order n and elements whose smallest eigenvalue has a high multiplicity. For n elliptic points A 1 , A 2 , …, A n , carried by the unit vectors a 1 , …, a n and spanning elliptic space E r−1 , the Gram matrix is symmetric, semipositive definite, and of rank r . B -matrices of order n ≡ 2r that have only two distinct eigenvalues with equal multiplicities r are called C -matrices. In view of the existence of a Hadamard matrix of order 92, it is interesting to know whether Paley's construction may be reversed to obtain a C -matrix of order 46.
287 citations
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01 Jan 1976
TL;DR: The problem of graph identification has been studied in the theory of permutation groups for a long time, see as mentioned in this paper for a discussion of the main points of the problem and an algorithm for graph identification.
Abstract: Some remarks about the problem of graph identification.- Motivation.- A construction of a stationary graph.- Properties of cells.- Properties of cellular algebras of rank greater than one.- Cellular algebras arising in the theory of permutation groups.- Some classes of cellular algebras.- Imprimitive cells and construction of factor-cells.- Construction of the quotient in the case of cellular algebras of rank greater than one.- On the structure of correct stationary graphs and cells having more than one normal subcell.- Properties of primitive cells.- Algebraic properties of cellular algebras.- Some modifications of stabilization.- Kernels and stability with respect to kernels.- Deep stabilization.- Examples of results using the stability of depth 1.- Some definitions and explanations about exhaustive search.- An algorithm of graph canonization.- A practical algorithm of graph canonization.- An algorithm of construction of strongly regular graphs.- Tables of strongly regular graphs with n vertices, 10?n?28.- Some properties of 25- and 26- families.
218 citations
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113 citations
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95 citations