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ON THE SECOND-ORDER REED–MULLER CODE1

by Ákos G. Horváth , 2002
"... Dedicated to the 75th birthday of István Reiman In this paper we shall give a recursion and a new explicite formula for some functions connected with the weight distribution of the second-order Reed–Muller code. We define some new subcodes of it and determine their information rates, respectively. ..."
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Dedicated to the 75th birthday of István Reiman In this paper we shall give a recursion and a new explicite formula for some functions connected with the weight distribution of the second-order Reed–Muller code. We define some new subcodes of it and determine their information rates, respectively.

List decoding of second order Reed-Muller codes

by Grigory Kabatiansky - In : Proceedings of the Eighteen International Symposium of Communication Theory and Applications, Ambleside , 2005
"... A new list decoding algorithms for second order Reed-Muller codes RM(2,m) of length n = 2m correcting far beyond minimal distance is proposed. In order to prove polynomial complexity of the algorithm we derive an improvement of well known Johnson bound. Key words: list decoding, complexity, Reed-Mul ..."
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A new list decoding algorithms for second order Reed-Muller codes RM(2,m) of length n = 2m correcting far beyond minimal distance is proposed. In order to prove polynomial complexity of the algorithm we derive an improvement of well known Johnson bound. Key words: list decoding, complexity, Reed-Muller

A fast reconstruction algorithm for deterministic compressive sensing using second order Reed-Muller codes

by S. D. Howard, A. R. Calderbank, S. J. Searle - Conference on Information Sciences and Systems (CISS), Princeton, ISBN: 978-1-4244-2246-3, pp: 11 - 15 , 2008
"... Abstract—This paper proposes a deterministic compressed sensing matrix that comes by design with a very fast reconstruction algorithm, in the sense that its complexity depends only on the number of measurements n and not on the signal dimension N. The matrix construction is based on the second order ..."
Abstract - Cited by 31 (5 self) - Add to MetaCart
order Reed-Muller codes and associated functions. This matrix does not have RIP uniformly with respect to all k-sparse vectors, but it acts as a near isometry on k-sparse vectors with very high probability. I.

List Decoding of Second Order Reed-Muller Codes and its Covering Radius Implications

by unknown authors
"... Abstract. An improvement of the Kabatiansky-Tavernier list decoding algorithm for the second order Reed-Muller code RM(2,m), of length n = 2m, and a better evaluation of its complexity (with a gain of a factor n) are proposed. We give a new bound for this complexity and conjecture a better bound whi ..."
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Abstract. An improvement of the Kabatiansky-Tavernier list decoding algorithm for the second order Reed-Muller code RM(2,m), of length n = 2m, and a better evaluation of its complexity (with a gain of a factor n) are proposed. We give a new bound for this complexity and conjecture a better bound

A linear construction for certain Kerdock and Preparata codes

by A. R. Calderbank, A. R. Hammons, P. Vijay Kumar, N. J. A. Sloane, Patrick Sol É - Bull. Amer. Math. Soc , 1993
"... codes are shown to be linear over Z4, the integers mod 4. The Kerdock and Preparata codes are duals over Z4, and the Nordstrom-Robinson code is self-dual. All these codes are just extended cyclic codes over Z4. This provides a simple definition for these codes and explains why their Hamming weight d ..."
Abstract - Cited by 15 (3 self) - Add to MetaCart
distributions are dual to each other. First- and second-order Reed-Muller codes are also linear codes over Z4, but Hamming codes in general are not, nor is the Golay code. 1.

The Z_4-linearity of Kerdock, Preparata, Goethals, and related codes

by A. Roger Hammons, Jr., P. Vijay Kumar, A. R. Calderbank, N. J. A. Sloane, Patrick Solé , 2001
"... 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 ..."
Abstract - Cited by 178 (15 self) - Add to MetaCart
are extended cyclic codes over ¡ 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

Low-degree tests at large distances

by Alex Samorodnitsky , 2006
"... We define tests of boolean functions which distinguish between linear (or quadratic)polynomials, and functions which are very far, in an appropriate sense, from these polynomials. The tests have optimal or nearly optimal trade-offs between soundness and the number of queries. In particular, we show ..."
Abstract - Cited by 48 (2 self) - Add to MetaCart
coding theory application. It is possible to estimate efficiently the distance from the second-order Reed-Muller code on inputs lying far beyond its list-decoding radius.

Coding of short control frames for UMTS mobile communication

by Wael Adi, Faisal Kriaa
"... Coding of short Control frames have gained more attention on the field of mobile communication. In recent mobile systems, very different services, channel data rates and formats are in use, especially in the 3 rd Generation Mobile Phone UMTS/3GPP. For this reason many transport channels are introduc ..."
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for the composition of the traffic data. Therefore they have to be carefully protected. We investigate in this paper the performance of a correction scheme using a block product code based on b-Adjacent code and compare the results in terms of frame error rate with those obtained by using a shortened second order

Best Quadratic Approximations of Cubic Boolean Functions

by Nicholas Kolokotronis, Konstantinos Limniotis, Nicholas Kalouptsidis , 2007
"... The problem of computing best low order approximations of Boolean functions is treated in this paper. We focus on the case of best quadratic approximations of a wide class of cubic functions of arbitrary number of variables, and provide formulas for their efficient calculation. Our methodology is ..."
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functions, and cubic functions that achieve the maximum possible nonquadricity are determined, leading to a lower bound for the covering radius of second order Reed-Muller code R(2, n) in R(3, n).

Intersections of Golay codes with higher order Kerdock codes

by Mohammed Abouzaid, Nick Gurski , 1999
"... Golay codewords are useful for transmission schemes such as OFDM, and [MacWilliams/Sloane] demonstrated that these codewords are bent functions contained in second order Reed-Muller codes. This paper investigates the intersection between these codes and Kerdock-like codes. 1 1 Introduction 1.1 Pur ..."
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Golay codewords are useful for transmission schemes such as OFDM, and [MacWilliams/Sloane] demonstrated that these codewords are bent functions contained in second order Reed-Muller codes. This paper investigates the intersection between these codes and Kerdock-like codes. 1 1 Introduction 1
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