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of
24
Generalized ReedMuller Codes and Power Control in OFDM Modulation
 IEEE Trans. Inform. Theory
, 2000
"... encoding, power, ..."
Algebraic Property Testing: The Role of Invariance
, 2007
"... We argue that the symmetries of a property being tested play a central role in property testing. We support this assertion in the context of algebraic functions, by examining properties of functions mapping a vector space K n over a field K to a subfield F. We consider Flinear properties that are i ..."
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Cited by 35 (16 self)
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We argue that the symmetries of a property being tested play a central role in property testing. We support this assertion in the context of algebraic functions, by examining properties of functions mapping a vector space K n over a field K to a subfield F. We consider Flinear properties that are invariant under linear transformations of the domain and prove that an O(1)local “characterization ” is a necessary and sufficient condition for O(1)local testability when K  = O(1). (A local characterization of a property is a definition of a property in terms of local constraints satisfied by functions exhibiting a property.) For the subclass of properties that are invariant under affine transformations of the domain, we prove that the existence of a single O(1)local constraint implies O(1)local testability. These results generalize and extend the class of algebraic properties, most notably linearity and lowdegreeness, that were previously known to be testable. In particular, the extensions include properties satisfied by functions of degree linear in n that turn out to be O(1)locally testable. Our results are proved by introducing a new notion that we term “formal characterizations”. Roughly this corresponds to characterizations that are given by a single local constraint and its permutations under linear transformations of the domain. Our main testing result shows that local formal characterizations
Which Linear Codes Are AlgebraicGeometric?
 IEEE Trans. Inform. Theory
, 1991
"... An infinite series of curves is constructed in order to show that all linear codes can be obtained from curves using Goppa's construction. If one imposes conditions on the degree of the divisor used, then we derive criteria for linear codes to be algebraicgeometric. In particular, we investigate th ..."
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Cited by 9 (3 self)
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An infinite series of curves is constructed in order to show that all linear codes can be obtained from curves using Goppa's construction. If one imposes conditions on the degree of the divisor used, then we derive criteria for linear codes to be algebraicgeometric. In particular, we investigate the family of qary Hamming codes, and prove that only those with redundancy one or two, and the binary [7; 4; 3] code are algebraicgeometric in this sense. For these codes we explicitly give a curve, rational points and a divisor. We prove that this triple is in a certain sense unique in the case of the [7; 4; 3] code. Key words: algebraicgeometric codes, algebraic curves, divisors, generalized Goppa codes, geometric Goppa codes. I. Introduction Since the early papers by Goppa [5],[6],[7], [8], algebraicgeometric codes have been in the spotlight of coding theoretic research for about a decade. As is wellknown, numerous exciting results have been achieved using Goppa's construction of li...
Geometric Codes over Fields of Odd Prime Power Order
 Congr. Numer
, 1999
"... We obtain improved bounds for the minimum weight of the dual codes associated with the codes from finite geometries in the case of odd order, and some results that apply also to the dual codes of nondesarguesian planes of odd order. ..."
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Cited by 8 (7 self)
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We obtain improved bounds for the minimum weight of the dual codes associated with the codes from finite geometries in the case of odd order, and some results that apply also to the dual codes of nondesarguesian planes of odd order.
The shift bound for cyclic, ReedMuller and geometric Goppa codes
 APPEARED IN ARITHMETIC, GEOMETRY AND CODING THEORY 4, LUMINY
, 1996
"... We give a generalization of the shift bound on the minimum distance for cyclic codes which applies to ReedMuller and algebraicgeometric codes. The number of errors one can correct by majority coset decoding is up to half the shift bound. ..."
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Cited by 8 (3 self)
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We give a generalization of the shift bound on the minimum distance for cyclic codes which applies to ReedMuller and algebraicgeometric codes. The number of errors one can correct by majority coset decoding is up to half the shift bound.
Bases of MinimumWeight Vectors for Codes from Designs
 Finite Fields Appl
, 1997
"... An explicit basis of incidence vectors for the pary code of the design of points and hyperplanes of the affine geometry AGm (F p ) for any prime p and any integer m # 2 is obtained, which, as a corollary, gives a new elementary proof that this code is a generalized ReedMuller code. In the proof a ..."
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Cited by 8 (5 self)
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An explicit basis of incidence vectors for the pary code of the design of points and hyperplanes of the affine geometry AGm (F p ) for any prime p and any integer m # 2 is obtained, which, as a corollary, gives a new elementary proof that this code is a generalized ReedMuller code. In the proof a class of nonsingular matrices related to Vandermonde matrices is introduced.
Generalized Hamming weights of qary ReedMuller codes
 IEEE Trans. Inform. Theory
, 1998
"... Abstract The order bound on generalized Hamming weights is introduced in a general setting of codes on varieties which comprises both the one point geometric Goppa codes as the qary ReedMuller codes. For the latter codes it is shown that this bound is sharp and that they satisfy the double chain c ..."
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Cited by 7 (1 self)
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Abstract The order bound on generalized Hamming weights is introduced in a general setting of codes on varieties which comprises both the one point geometric Goppa codes as the qary ReedMuller codes. For the latter codes it is shown that this bound is sharp and that they satisfy the double chain condition. 1
Minimumweight codewords as generators of generalized ReedMuller codes
, 2000
"... We establish the range of values of #, where 0 1), for which the generalized ReedMuller code RF q (#, m) of length q over the field F q of order q is spanned by its minimumweight vectors. ..."
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Cited by 7 (1 self)
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We establish the range of values of #, where 0 1), for which the generalized ReedMuller code RF q (#, m) of length q over the field F q of order q is spanned by its minimumweight vectors.
Simple MAP decoding of firstorder ReedMuller and Hamming codes
 IEEE Trans. Inf. Theory
, 2004
"... Abstract—A maximum a posteriori (MAP) probability decoder of a block code minimizes the probability of error for each transmitted symbol separately. The standard way of implementing MAP decoding of a linear code is the Bahl–Cocke–Jelinek–Raviv (BCJR) algorithm, which is based on a trellis representa ..."
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Cited by 6 (0 self)
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Abstract—A maximum a posteriori (MAP) probability decoder of a block code minimizes the probability of error for each transmitted symbol separately. The standard way of implementing MAP decoding of a linear code is the Bahl–Cocke–Jelinek–Raviv (BCJR) algorithm, which is based on a trellis representation of the code. The complexity of the BCJR algorithm for the firstorder Reed–Muller (RM1) codes and Hamming codes is proportional to, where is the code’s length. In this correspondence, we present new MAP decoding algorithms for binary and nonbinary RM1 and Hamming codes. The proposed algorithms have complexities proportional to �� � , where is the alphabet size. In particular, for the binary codes this yields complexity of order �� �. Index Terms—Hamming codes, maximum a posteriori (MAP) decoding, Reed–Muller codes. I.