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Cryptography
Published in Paul L. Goethals, Natalie M. Scala, Daniel T. Bennett, Mathematics in Cyber Research, 2022
Gretchen L. Matthews, Aidan W. Murphy
The minimum distance d of the code C equals the minimum number of linearly dependent columns of a parity-check matrix H for C; that is, d=min{l:{Colj1H,…,ColjlH}is linearly dependent}.
DTN Coding
Published in Aloizio Pereira da Silva, Scott Burleigh, Katia Obraczka, Delay and Disruption Tolerant Networks, 2019
Marius Feldmann, Felix Walter, Tomaso de Cola, Gianluigi Liva
The generator matrix is in systematic form, which readily yields the parity-check matrix H=(111010011010101011001)
Mathematical Preliminaries
Published in Frank Y. Shih, Digital Watermarking and Steganography: Fundamentals and Techniques, 2017
An example of error correction is shown in Figure 3.16. By multiplying the obtained code stream and the parity check matrix and taking modulo-2, the syndrome vector can be obtained to indicate whether an error has occurred and, if so, for which code word bit. In Figure 3.16a, the syndrome vector is (0,0,0), indicating there is no error. If the original 7-bit stream was 1111111, then a single bit (i.e., the second bit) error occurred during transmission, causing the received 7-bit stream 1011111. In Figure 3.16b, the syndrome vector is (1,0,1), corresponding to the second row of the parity check matrix. Therefore, the recipient will know that the second bit of the obtained bitstream is incorrect.
A Quantum Resistant Chameleon Hashing and Signature Scheme
Published in IETE Journal of Research, 2022
A linear code C is a k-dimensional linear subspace of and all the elements of C are called codewords. The weight of an element x in is the number of nonzero coordinates of x . The minimum distanced of a linear code is the minimum weight of any nonzero element of the code. A code is specified either by a generator matrix G or by a parity check matrix H. A generator matrix of a linear code C is a matrix G whose row vectors form a basis for the vector subspace C, i.e.,. A parity-check matrixH of a linear code C is an matrix whose row vectors form a basis for the orthogonal complement of the vector subspace C. Hence . The syndrome of a word by H is a vector defined by . If the error correcting capability of the code is then for any syndrome s, there is at most one word of weight , such that . If there exists a word x corresponding to a vector , then s is said to be t-decodable (or simply decodable) in the code defined by H. Given a random syndrome s, finding whether s is t-decodable in the code defined by a random matrix H has been proved to be NP-complete [16]. Decoding can be easily performed for more structured codes.