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## On Correctable Errors of Binary Linear Codes (2010)

Citations: | 1 - 0 self |

### Citations

2656 | The Theory of Error-Correcting Codes - MacWilliams, Sloane - 1977 |

1365 | Low-Density Parity-Check Codes. - Gallager - 1963 |

281 | An Introduction to Probability Theory and its Applications, 3rd Edition - Feller - 1957 |

122 | Boolean Functions for Cryptography and Error Correcting Codes, Boolean Models and Methods - Carlet - 2010 |

76 | Bounds on the decoding error probability of binary linear codes via their spectra,” - Poltyrev - 1994 |

58 | Random codes: Minimum distances and error exponents
- Barg, Forney
- 2002
(Show Context)
Citation Context ...ODES 2541 for any , we have that . Since is a monotonically increasing function of , we have that for typical random linear codes. The weight distribution of typical random linear codes is studied in =-=[16]-=-. Let with be the number of codewords of weight for typical random linear codes. It is shown that , which is a monotonically increasing function of for . Thus, we have that for . From the above result... |

40 |
Weight distributions of the cosets of the (32; 6) Reed-Muller code
- Berlekamp, Welch
- 1972
(Show Context)
Citation Context ...ibution of the coset leaders represents the number of correctable errors for each weight. Considering this fact, the error correction capabilities for certain specific codes are completely determined =-=[1]-=-–[4]. For general linear codes, some bounds on the number of correctable errors were presented in [5]–[7]. Although the first-order Reed–Muller codes have a simple structure, the exact number of corre... |

35 | Minimal vectors in linear codes,
- Ashikhmin, Barg
- 1998
(Show Context)
Citation Context ...rial set for . Since every larger half is an uncorrectable error, we have the relation A codeword is called minimal if for implies . Basic properties and applications of minimal codewords are seen in =-=[12]-=-. Let denote the set of minimal codewords in . Then we have the following property [6, Corollary 5]: is a trial set for (4) then is also a trial set for (5) It follows from the above fact that the set... |

7 | The coset distribution of triple-error-correcting binary primitive BCH codes - Charpin, Helleseth, et al. - 2006 |

7 |
The Newton radius of codes
- Helleseth, Kløve
- 1997
(Show Context)
Citation Context ... this fact, the error correction capabilities for certain specific codes are completely determined [1]–[4]. For general linear codes, some bounds on the number of correctable errors were presented in =-=[5]-=-–[7]. Although the first-order Reed–Muller codes have a simple structure, the exact number of correctable errors for them was known only for weight [8]. Determining the number of correctable errors of... |

6 |
Levenshtein: Error correction capability of binary linear codes
- Helleseth, Kløve, et al.
- 2005
(Show Context)
Citation Context ...ucture is useful for error analysis since the correctable (and uncorrectable) errors are characterized by the maximal correctable (and minimal uncorrectable) errors. Helleseth, Kløve, and Levenshtein =-=[6]-=- analyzed an asymptotic error correctability of binary linear codes using the monotone error structure, and introduced two useful notions, larger half and trial set, which characterizes the minimal un... |

6 | distribution of Boolean functions with nonlinearity ≤ 2n−2
- Wu, “On
- 1998
(Show Context)
Citation Context ...e number of correctable errors were presented in [5]–[7]. Although the first-order Reed–Muller codes have a simple structure, the exact number of correctable errors for them was known only for weight =-=[8]-=-. Determining the number of correctable errors of weight for the first-order Reed–Muller codes is equivalent to determining the number of Boolean functions with nonlinearity , and the nonlinearity of ... |

3 | Weight distributions of cosets of two-error-correcting binary BCH codes, extended or not - Charpin - 1994 |

3 |
On cryptographic properties of the cosets of
- Canteaut, Carlet, et al.
- 2001
(Show Context)
Citation Context ... an important criterion for cryptographic system, block ciphers and stream ciphers. Many studies have been conducted on the nonlinearity of Boolean functions in cryptography. For further details, see =-=[17]-=-, [9] and references therein. For an integer is defined recursively as . for for Proof: First we observe that . Let and be codewords in . Then where denotes the concatenation of and , and . Since all ... |

2 |
Weight distribution of the coset leaders of some Reed–Muller codes and BCH codes
- Maeda, Fujiwara
- 2001
(Show Context)
Citation Context ...ion of the coset leaders represents the number of correctable errors for each weight. Considering this fact, the error correction capabilities for certain specific codes are completely determined [1]–=-=[4]-=-. For general linear codes, some bounds on the number of correctable errors were presented in [5]–[7]. Although the first-order Reed–Muller codes have a simple structure, the exact number of correctab... |

1 |
Threshold effects in codes,” presented at the Algebraic Coding
- Zémor
- 1993
(Show Context)
Citation Context ...Consider and with . The monotone structure is the following property: If is a correctable error, then is also correctable. If is uncorrectable, then is also uncorrectable. Using this structure, Zémor =-=[11]-=-, [18] elucidated that the error probability after the maximum likelihood decoding over the binary symmetric channels displays a threshold behavior. Helleseth et al. [6] studied this structure and int... |

1 |
Kenji Yasunaga received the B.E. degree in information and computer sciences in 2003 and the M.Sc. and Ph.D. degrees in information science and technology in 2005 and 2008, from Osaka University, Japan. His research interests are in coding theory, cryptog
- E, Ph
- 1994
(Show Context)
Citation Context ...er and with . The monotone structure is the following property: If is a correctable error, then is also correctable. If is uncorrectable, then is also uncorrectable. Using this structure, Zémor [11], =-=[18]-=- elucidated that the error probability after the maximum likelihood decoding over the binary symmetric channels displays a threshold behavior. Helleseth et al. [6] studied this structure and introduce... |