## Linearity testing in characteristic two (1996)

### Cached

### Download Links

Venue: | IEEE Transactions on Information Theory |

Citations: | 53 - 7 self |

### BibTeX

@ARTICLE{Bellare96linearitytesting,

author = {M. Bellare and M. Kiwi and M. Sudan},

title = {Linearity testing in characteristic two},

journal = {IEEE Transactions on Information Theory},

year = {1996}

}

### Years of Citing Articles

### OpenURL

### Abstract

The case we are interested in is when the underlying groups are G=GF(2)n and H=GF(2). In this case the collection of linear functions describe a Hadamard code of block length 2n and for an arbitrary function f mapping GF(2)n to GF(2) the distance Dist(f) measures its distance to a Hadamard code (normalized so as to be a real number between 0 and 1). The quantity Err(f) is a parameter that is "easy to measure " and linearity testing studies the relationship of this parameter to the distance of f. The code and corresponding test are used in the construction of efficient probabilistically checkable proofs and thence in the derivation of hardness of approximation results. In this context, improved analyses translate into better non-approximability results. However, while several analyses of the relation of Err(f) to Dist(f) are known, none is tight.

### Citations

2052 | The theory of error-correcting codes - MacWilliams, Sloane - 1977 |

1707 | The Probabilistic Method
- Alon, Spencer
- 1992
(Show Context)
Citation Context ...esent a description of the relationship between Err(f ) and Dist(f ) which is nearly complete in all its aspects, and entirely complete (i.e. tight) in some. In particular we present functions L; U : =-=[0; 1]-=- ! [0; 1] such that for all x 2 [0; 1] we have L(x) ^ Err(f ) ^ U (x) whenever Dist(f ) = x, with the upper bound being tight on the whole range, and the lower bound tight on a large part of the range... |

406 | Nondeterministic exponential time has two-prover interactive protocols
- Babai, Fortnow, et al.
- 1991
(Show Context)
Citation Context ...nearity testing. In the low total degree testing problem we are asked to determine whether f is close to some polynomial p of total degree d in its n variables.5 Multi-linearity tests were studied by =-=[4, 11]-=-. Low individual degree tests were studied by [3, 5, 12, 19]. Total degree tests were studied by [2, 13, 14, 20]. What we are looking at, namely linearity testing over GF(2), is a variant of the total... |

366 | Probabilistic checking of proofs: a new characterization of NP - Arora, Safra - 1998 |

349 | Self-testing/correcting with applications to numerical problems
- Blum, Luby, et al.
- 1993
(Show Context)
Citation Context ...R G [ f (u)6=g(u) ] -- (relative) distance between f; g: G ! H Dist(f ) def = minf Dist(f; g) : g 2 Lin(G; H) g -- Distance of f to its closest linear function. The BLR Test. Blum, Luby and Rubinfeld =-=[9]-=- suggest a probabilistic method to "test" if a function f is really a linear function. This test, henceforth referred to as the BLR test, is the following [9]-- Given a function f : G ! H, pick u; v 2... |

256 | Checking computations in polylogarithmic time
- Babai, Fortnow, et al.
- 1991
(Show Context)
Citation Context ...lem we are asked to determine whether f is close to some polynomial p of total degree d in its n variables.5 Multi-linearity tests were studied by [4, 11]. Low individual degree tests were studied by =-=[3, 5, 12, 19]-=-. Total degree tests were studied by [2, 13, 14, 20]. What we are looking at, namely linearity testing over GF(2), is a variant of the total degree testing problem in which the degree is d = 1, F is s... |

206 | Proof verification and intractability of approximation problems - Arora, Lund, et al. - 1998 |

180 | Approximating clique is almost npcomplete - Feige, Goldwasser, et al. - 1991 |

169 | Efficient probabilistically checkable proofs: Applications to approximation
- Bellare, Goldwasser, et al.
- 1993
(Show Context)
Citation Context ... be significant in this example and an important parameter that emerges in the study of linearity testing is how low Rej G;H(x) can be for x * 14 . In this paper we call this parameter, identified in =-=[2, 6, 7, 8]-=-, the knee of the curve. Formally: Knee G;H def = minf Rej(x) : x * 14 g : 1.2 Error detection in Hadamard codes In this paper we look at the performance of the BLR test when the underlying groups are... |

113 | Improved non-approximability results - Bellare, Sudan - 1994 |

77 | Selftesting/correcting for polynomials and for approximate functions
- Gemmell, Lipton, et al.
- 1991
(Show Context)
Citation Context ... some polynomial p of total degree d in its n variables.5 Multi-linearity tests were studied by [4, 11]. Low individual degree tests were studied by [3, 5, 12, 19]. Total degree tests were studied by =-=[2, 13, 14, 20]-=-. What we are looking at, namely linearity testing over GF(2), is a variant of the total degree testing problem in which the degree is d = 1, F is set to GF(2), and the constant term of the polynomial... |

42 | Some improvements to total degree tests
- Friedl, Sudan
- 1995
(Show Context)
Citation Context ... some polynomial p of total degree d in its n variables.5 Multi-linearity tests were studied by [4, 11]. Low individual degree tests were studied by [3, 5, 12, 19]. Total degree tests were studied by =-=[2, 13, 14, 20]-=-. What we are looking at, namely linearity testing over GF(2), is a variant of the total degree testing problem in which the degree is d = 1, F is set to GF(2), and the constant term of the polynomial... |

25 | Selftesting /correcting for polynomials and for approximate functions - Gemmell, Lipton, et al. - 1991 |

22 |
Low-degree tests
- HÁTSÁGI, SHEN
- 1994
(Show Context)
Citation Context ...lem we are asked to determine whether f is close to some polynomial p of total degree d in its n variables.5 Multi-linearity tests were studied by [4, 11]. Low individual degree tests were studied by =-=[3, 5, 12, 19]-=-. Total degree tests were studied by [2, 13, 14, 20]. What we are looking at, namely linearity testing over GF(2), is a variant of the total degree testing problem in which the degree is d = 1, F is s... |

21 | Proof veri and intractability of approximation problems - Arora, Lund, et al. - 1992 |

19 |
Robust Characterizations of Polynomials and Their Applications to Program Testing
- Rubinfeld, Sudan
- 1996
(Show Context)
Citation Context ... some polynomial p of total degree d in its n variables.5 Multi-linearity tests were studied by [4, 11]. Low individual degree tests were studied by [3, 5, 12, 19]. Total degree tests were studied by =-=[2, 13, 14, 20]-=-. What we are looking at, namely linearity testing over GF(2), is a variant of the total degree testing problem in which the degree is d = 1, F is set to GF(2), and the constant term of the polynomial... |

19 | Ecient probabilistically checkable proofs - Bellare, Goldwasser, et al. - 1993 |

15 | Probabilistically Checkable Proofs and the Testing of Hadamard-like Codes
- Kiwi
- 1996
(Show Context)
Citation Context ...[7] use all these different testers, but, as we explained, the final non-approximability factors obtained can be expressed only in terms of the shape of the linearity testing curve. Recent work. Kiwi =-=[16]-=- provides improved analysis for the linearity testing problem over all finite fields. He obtains this result by providing another new interpretation of the linearity testing problem, this time by rela... |

12 |
Free bits and non-approximability
- Bellare, Goldreich, et al.
- 1995
(Show Context)
Citation Context ...rd problems. Usage of the linearity test in the construction of efficient PCPs, and thence in the derivation of hardness of approximability results for MaxSNP problems, begins in [2] and continues in =-=[6, 8, 7]-=-. In the first three cases, it is used to test the Hadamard code; in the last case, to test a different code called the long code. In all cases the underlying problem is the one we have considered abo... |

11 |
Nearly Linear Sized Holographic Proofs
- Polishchuk, Spielman
- 1994
(Show Context)
Citation Context ...lem we are asked to determine whether f is close to some polynomial p of total degree d in its n variables.5 Multi-linearity tests were studied by [4, 11]. Low individual degree tests were studied by =-=[3, 5, 12, 19]-=-. Total degree tests were studied by [2, 13, 14, 20]. What we are looking at, namely linearity testing over GF(2), is a variant of the total degree testing problem in which the degree is d = 1, F is s... |

4 |
Private communication
- Kleitman
- 1994
(Show Context)
Citation Context ...tasB \ThetasC such that u+v+w = 0, is maximized when A; B; C are the lexicographically smallest jAj; jBj; jCj elements of F n respectively. The following lemma, independently proved by D. J. Kleitman =-=[17]-=-, gives a precise statement of the above discussed fact. For convenience we introduce the following notation: for every nonnegative integer n and A; B; C ` F n let \Phi n(A; B; C) = f (u; v; w) 2 A\Th... |

2 | Low-degree testing - agi, Shen - 1994 |

1 | Testing of the long code and hardness for clique. To appear - astad - 1996 |

1 |
Testing of the long code and hardness for clique. To appear
- Hastad
- 1996
(Show Context)
Citation Context ... new interpretation of the linearity testing problem, this time by relating it to a weight enumeration problem of a linear code studied as a function of the minimum distance of its dual code. H*astad =-=[15]-=- has shown a tester for a different code, namely the "long code" of [7], and an analysis for the test is again based on a Fourier Transform based approach. The analysis once again provides significant... |