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37
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 36 (15 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
Testing LowDegree Polynomials over Prime Fields
, 2006
"... We present an efficient randomized algorithm to test if a given function f: Fnp! Fp (where p is a prime) is a lowdegree polynomial. This gives a local test for Generalized ReedMuller codes over prime fields. For a given integer t and a given real ffl> 0, the algorithm queries f at O i 1ffl + ..."
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Cited by 33 (2 self)
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We present an efficient randomized algorithm to test if a given function f: Fnp! Fp (where p is a prime) is a lowdegree polynomial. This gives a local test for Generalized ReedMuller codes over prime fields. For a given integer t and a given real ffl> 0, the algorithm queries f at O i 1ffl + t * p 2tp1 +1j points to determine whether f can be described by a polynomial of degree at most t. If f is indeed a polynomial of degree at most t, our algorithm always accepts, and if f has a relative distance at least ffl from every degree t polynomial, then our algorithm rejects fwith probability at least 1 2. Our result is almost optimal since any such algorithm must query f on at least \Omega ( 1ffl + p
ReedMuller codes associated to projective algebraic varieties. In “Coding Theory and Algebraic Geomertry and Coding Theory
 Lecture Notes in Math
"... The classical generalized ReedMuller cedes introduced by Kasami, Lin and Peterson [5], and studied also by Delsarte, Goethals and Mac Williams [2], are defined over the affine space An(Fq) over the finite field Fq with q elements. Moreover Lachand [6], following Manin and Vladut [7], has considered ..."
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Cited by 13 (0 self)
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The classical generalized ReedMuller cedes introduced by Kasami, Lin and Peterson [5], and studied also by Delsarte, Goethals and Mac Williams [2], are defined over the affine space An(Fq) over the finite field Fq with q elements. Moreover Lachand [6], following Manin and Vladut [7], has considered projective ReedMuller codes, i.e. defined over the projective space pn(Fq). In this paper, the evaluation of the forms with coefficients in the finite field Fq is made on the points of a projective algebraic variety V over the projective space pn~q). Firstly, we consider the case where V is a quadric hypersurface, singular or not, Parabolic, Hyperbolic or Elliptic. Some results about he number of points in a (possibly degenerate) quadric and in the hyperplane s ctions are given, and also is given an upper bound of the number ofpoints in the intersection ftwo quadrics. In application of these results, we obtain ReedMuller codes of order 1 associated toquadrics with three weights and we give their parameters, as well as ReedMuller codes of order 2 with their parameters, Secondly, we take V as a hypersurface, which is the union of hyperplanes containing a linear variety of codimension 2 (these hypersurfaces reach the Serre bound). If V is of degree h, we give parameters ofReedMuller codes of order d < h, associated toV.
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 investiga ..."
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Cited by 11 (4 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...
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 11 (5 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.
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 9 (8 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.
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 9 (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
Decoding affine variety codes using Gröbner bases
 CODES CRYPTOGR
, 1998
"... We define a class of codes that we call affine variety codes. These codes are obtained by evaluating functions in the coordinate ring of an affine variety on all the Fqrational points of the variety. We show that one can, at least in theory, decode these codes up to half the true minimum distance ..."
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Cited by 8 (0 self)
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We define a class of codes that we call affine variety codes. These codes are obtained by evaluating functions in the coordinate ring of an affine variety on all the Fqrational points of the variety. We show that one can, at least in theory, decode these codes up to half the true minimum distance by using the theory of Gröbner bases. We extend results of A. B. Cooper and of X. Chen, I. S. Reed, T. Helleseth, and T. K. Truong.
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.