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Journal articles on the topic 'Hamming codes'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 April 2022

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1

Rifà, J., F. I. Solov’eva, and M. Villanueva. "Intersection of Hamming codes avoiding Hamming subcodes." Designs, Codes and Cryptography 62, no. 2 (2011): 209–23. http://dx.doi.org/10.1007/s10623-011-9506-0.

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2

Upadhyay, Rohitkumar R. "Hamming Codes: Error Reducing Techniques." International Journal for Research in Applied Science and Engineering Technology 9, no. 11 (2021): 1972–74. http://dx.doi.org/10.22214/ijraset.2021.39146.

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Abstract: Hamming codes for all intents and purposes are the first nontrivial family of error-correcting codes that can actually correct one error in a block of binary symbols, which literally is fairly significant. In this paper we definitely extend the notion of error correction to error-reduction and particularly present particularly several decoding methods with the particularly goal of improving the error-reducing capabilities of Hamming codes, which is quite significant. First, the error-reducing properties of Hamming codes with pretty standard decoding definitely are demonstrated and explored. We show a sort of lower bound on the definitely average number of errors present in a decoded message when two errors for the most part are introduced by the channel for for all intents and purposes general Hamming codes, which actually is quite significant. Other decoding algorithms are investigated experimentally, and it generally is definitely found that these algorithms for the most part improve the error reduction capabilities of Hamming codes beyond the aforementioned lower bound of for all intents and purposes standard decoding. Keywords: coding theory, hamming codes, hamming distance

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3

Barashko, A. S. "Polynomials generating Hamming codes." Ukrainian Mathematical Journal 45, no. 7 (1993): 987–92. http://dx.doi.org/10.1007/bf01057445.

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4

Bauer, Friedrich L. "Richard Hamming: Fehlerkorrigierende Codes." Informatik-Spektrum 30, no. 2 (2007): 95–99. http://dx.doi.org/10.1007/s00287-006-0135-3.

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5

Borges, Joaquim, Josep Rifà, and Victor Zinoviev. "Completely regular codes by concatenating Hamming codes." Advances in Mathematics of Communications 12, no. 2 (2018): 337–49. http://dx.doi.org/10.3934/amc.2018021.

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6

González-Sarabia, Manuel, Delio Jaramillo, and Rafael H. Villarreal. "On the generalized Hamming weights of certain Reed–Muller-type codes." Analele Universitatii "Ovidius" Constanta - Seria Matematica 28, no. 1 (2020): 205–17. http://dx.doi.org/10.2478/auom-2020-0014.

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AbstractThere is a nice combinatorial formula of P. Beelen and M. Datta for the r-th generalized Hamming weight of an a ne cartesian code. Using this combinatorial formula we give an easy to evaluate formula to compute the r-th generalized Hamming weight for a family of a ne cartesian codes. If 𝕏 is a set of projective points over a finite field we determine the basic parameters and the generalized Hamming weights of the Veronese type codes on 𝕏 and their dual codes in terms of the basic parameters and the generalized Hamming weights of the corresponding projective Reed–Muller-type codes on 𝕏 and their dual codes.

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7

Rifà, J., F. I. Solov’eva, and M. Villanueva. "Erratum to: Intersection of Hamming codes avoiding Hamming subcodes." Designs, Codes and Cryptography 74, no. 1 (2014): 283. http://dx.doi.org/10.1007/s10623-014-0011-0.

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8

Sarabia, Manuel González. "Some bounds for the relative generalized Hamming weights of some evaluation codes." Analele Universitatii "Ovidius" Constanta - Seria Matematica 24, no. 2 (2016): 261–70. http://dx.doi.org/10.1515/auom-2016-0041.

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Abstract In this paper we find some bounds for the relative generalized Hamming weights of some codes parameterized by a set of monomials of the same degree. Also we compare the relative generalized Hamming weights of the codes CX(d) and CX′(d) when X′ is embedded in X. We use these results to obtain some lower bounds for the relative generalized Hamming weights of the codes parameterized by the edges of any connected bipartite graph with bipartition (V1, V2) and where |V1| = n1, |V2| = n2, in terms of the relative generalized Hamming weights of the codes associated to the projective tori Tn₁-1 and Tn₂-1.

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9

HAWTIN, DANIEL R., NEIL I. GILLESPIE, and CHERYL E. PRAEGER. "ELUSIVE CODES IN HAMMING GRAPHS." Bulletin of the Australian Mathematical Society 88, no. 2 (2013): 286–96. http://dx.doi.org/10.1017/s0004972713000051.

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AbstractWe consider a code to be a subset of the vertex set of a Hamming graph. We examine elusive pairs, code-group pairs where the code is not determined by knowledge of its set of neighbours. We construct a new infinite family of elusive pairs, where the group in question acts transitively on the set of neighbours of the code. In these examples, the alphabet size always divides the length of the code. We show that there is no elusive pair for the smallest set of parameters that does not satisfy this condition. We also pose several questions regarding elusive pairs.

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10

Phelps, Kevin T., and Mike Levan. "Kernels of nonlinear Hamming codes." Designs, Codes and Cryptography 6, no. 3 (1995): 247–57. http://dx.doi.org/10.1007/bf01388478.

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11

Wood, Jay A. "Isometry groups of additive codes over finite fields." Journal of Algebra and Its Applications 17, no. 10 (2018): 1850198. http://dx.doi.org/10.1142/s0219498818501980.

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When [Formula: see text] is a linear code over a finite field [Formula: see text], every linear Hamming isometry of [Formula: see text] to itself is the restriction of a linear Hamming isometry of [Formula: see text] to itself, i.e. a monomial transformation. This is no longer the case for additive codes over non-prime fields. Every monomial transformation mapping [Formula: see text] to itself is an additive Hamming isometry, but there may exist additive Hamming isometries that are not monomial transformations.The monomial transformations mapping [Formula: see text] to itself form a group [Formula: see text], and the additive Hamming isometries form a larger group [Formula: see text]: [Formula: see text]. The main result says that these two subgroups can be as different as possible: for any two subgroups [Formula: see text], subject to some natural necessary conditions, there exists an additive code [Formula: see text] such that [Formula: see text] and [Formula: see text].

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12

Falcone, Giovanni, and Marco Pavone. "Binary Hamming codes and Boolean designs." Designs, Codes and Cryptography 89, no. 6 (2021): 1261–77. http://dx.doi.org/10.1007/s10623-021-00853-z.

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AbstractIn this paper we consider a finite-dimensional vector space $${\mathcal {P}}$$ P over the Galois field $${\text {GF}}(2),$$ GF ( 2 ) , and the family $${\mathcal {B}}_k$$ B k (respectively, $${\mathcal {B}}_k^*$$ B k ∗ ) of all the k-sets of elements of $$\mathcal {P}$$ P (respectively, of $${\mathcal {P}}^*= {\mathcal {P}} \setminus \{0\}$$ P ∗ = P \ { 0 } ) summing up to zero. We compute the parameters of the 3-design $$({\mathcal {P}},{\mathcal {B}}_k)$$ ( P , B k ) for any (necessarily even) k, and of the 2-design $$({\mathcal {P}}^{*},{\mathcal {B}}_k^{*})$$ ( P ∗ , B k ∗ ) for any k. Also, we find a new proof for the weight distribution of the binary Hamming code. Moreover, we find the automorphism groups of the above designs by characterizing the permutations of $${\mathcal {P}}$$ P , respectively of $${\mathcal {P}}^*$$ P ∗ , that induce permutations of $${\mathcal {B}}_k$$ B k , respectively of $${\mathcal {B}}_k^*.$$ B k ∗ . In particular, this allows one to relax the definitions of the permutation automorphism groups of the binary Hamming code and of the extended binary Hamming code as the groups of permutations that preserve just the codewords of a given Hamming weight.

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13

Godlewski, Philippe. "Wom-codes construits à partir des codes de hamming." Discrete Mathematics 65, no. 3 (1987): 237–43. http://dx.doi.org/10.1016/0012-365x(87)90055-0.

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14

Hu, Wan Bao, and Xiao Hong Fang. "Generalized Hamming Weights of Cyclic Codes from Algebraic Function Fields." Advanced Materials Research 756-759 (September 2013): 2588–92. http://dx.doi.org/10.4028/www.scientific.net/amr.756-759.2588.

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The generalized Hamming weights of a linear code are fundamental code parameters related to the minimal overlap structures of the subcodes. In this paper, some results are presented on the estimates of generalized Hamming weights of cyclic codes, which connect their corresponding trace codes.

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15

Li, Ruihu, Luobin Guo, and Zongben Xu. "Entanglement-assisted quantum codes achieving the quantum singleton bound but violating the quantum hamming bound." Quantum Information and Computation 14, no. 13&14 (2014): 1107–16. http://dx.doi.org/10.26421/qic14.13-14-4.

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We give an infinite family of degenerate entanglement-assisted quantum error-correcting codes (EAQECCs) which violate the EA-quantum Hamming bound for non-degenerate EAQECCs and achieve the EA-quantum Singleton bound, thereby proving that the EA-quantum Hamming bound does not asymptotically hold for degenerate EAQECCs. Unlike the previously known quantum error-correcting codes that violate the quantum Hamming bound by exploiting maximally entangled pairs of qubits, our codes do not require local unitary operations on the entangled auxiliary qubits during encoding. The degenerate EAQECCs we present are constructed from classical error-correcting codes with poor minimum distances, which implies that, unlike the majority of known EAQECCs with large minimum distances, our EAQECCs take more advantage of degeneracy and rely less on the error correction capabilities of classical codes.

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16

Tromp, John, Louxin Zhang, and Ying Zhao. "Small weight bases for hamming codes." Theoretical Computer Science 181, no. 2 (1997): 337–45. http://dx.doi.org/10.1016/s0304-3975(96)00278-2.

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17

Wei, V. K. "Generalized Hamming weights for linear codes." IEEE Transactions on Information Theory 37, no. 5 (1991): 1412–18. http://dx.doi.org/10.1109/18.133259.

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18

Helleseth, T., T. Klove, and O. Ytrehus. "Generalized Hamming weights of linear codes." IEEE Transactions on Information Theory 38, no. 3 (1992): 1133–40. http://dx.doi.org/10.1109/18.135655.

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19

Stichtenoth, H., and C. Voss. "Generalized Hamming weights of trace codes." IEEE Transactions on Information Theory 40, no. 2 (1994): 554–58. http://dx.doi.org/10.1109/18.312185.

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20

Ericson, T., and V. I. Levenshtein. "Superimposed codes in the Hamming space." IEEE Transactions on Information Theory 40, no. 6 (1994): 1882–93. http://dx.doi.org/10.1109/18.340463.

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21

Giudici, Michael, and Cheryl E. Praeger. "Completely Transitive Codes in Hamming Graphs." European Journal of Combinatorics 20, no. 7 (1999): 647–62. http://dx.doi.org/10.1006/eujc.1999.0313.

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22

Munuera, Carlos. "Hamming codes for wet paper steganography." Designs, Codes and Cryptography 76, no. 1 (2014): 101–11. http://dx.doi.org/10.1007/s10623-014-9998-5.

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23

Ding, Cunsheng, and Jing Yang. "Hamming weights in irreducible cyclic codes." Discrete Mathematics 313, no. 4 (2013): 434–46. http://dx.doi.org/10.1016/j.disc.2012.11.009.

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24

Fish, W., J. D. Key, E. Mwambene, and B. G. Rodrigues. "Hamming graphs and special LCD codes." Journal of Applied Mathematics and Computing 61, no. 1-2 (2019): 461–79. http://dx.doi.org/10.1007/s12190-019-01259-w.

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25

Podemski, Robert, Witold Holubowicz, Claude Berrou, and Gerard Battail. "Hamming distance spectra of turbo-codes." Annales Des Télécommunications 50, no. 9-10 (1995): 790–97. http://dx.doi.org/10.1007/bf02997783.

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26

Hawtin, Daniel R. "s-Elusive codes in Hamming graphs." Designs, Codes and Cryptography 89, no. 6 (2021): 1211–20. http://dx.doi.org/10.1007/s10623-021-00868-6.

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27

Hillier, Caleb, and Vipin Balyan. "Error Detection and Correction On-Board Nanosatellites Using Hamming Codes." Journal of Electrical and Computer Engineering 2019 (February 10, 2019): 1–15. http://dx.doi.org/10.1155/2019/3905094.

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The field of nanosatellites is constantly evolving and growing at a very fast speed. This creates a growing demand for more advanced and reliable EDAC systems that are capable of protecting all memory aspects of satellites. The Hamming code was identified as a suitable EDAC scheme for the prevention of single event effects on-board a nanosatellite in LEO. In this paper, three variations of Hamming codes are tested both in Matlab and VHDL. The most effective version was Hamming [16, 11, 4]2. This code guarantees single-error correction and double-error detection. All developed Hamming codes are suited for FPGA implementation, for which they are tested thoroughly using simulation software and optimized.

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28

Ivanov, Aleksandr I., and Andrey G. Bannykh. "Rapid Estimation of the Entropy of Long Codes with Dependent Bits on Low-Power, Low-Bit Microcontrollers (Review of Literature on Reducing the Dimension of a Problem)." Engineering Technologies and Systems 30, no. 2 (2020): 300–312. http://dx.doi.org/10.15507/2658-4123.030.202002.300-312.

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Introduction. The aim of the work is to reduce the requirements for bit depth and processor performance of a trusted computing environment when estimating the entropy of long codes with dependent bits. Materials and Methods. Testing procedures recommended by the Russia national standards are used. The transition from the analysis of ordinary long codes to Hamming distances between random Alien codes and the Own image code is used. Results. It is shown that the transition to the presentation of data by the normal distribution law in the space of Hamming distances makes the relationship between mathematical expectation and entropy almost linear. Low-bit tables are constructed that relate the first statistical moments of the distribution of Hamming distances to the entropy of long codes. In calculations, the correlation index of the digits of the studied codes can vary widely. Discussion and Conclusion. The calculation of the mathematical expectation and standard deviation is easily feasible on low-discharge low-power microcontrollers. The use of the synthesized tables makes it possible to pass easily from the lower statistical moments of the Hamming distances to the entropy of long codes. The task of calculating entropy is accelerated many times in comparison with Shannon’s procedures and becomes feasible on cheap low-bit processors.

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29

Alfaro, Ricardo, and Andrei V. Kelarev. "On cyclic codes in incidence rings." Studia Scientiarum Mathematicarum Hungarica 43, no. 1 (2006): 69–77. http://dx.doi.org/10.1556/sscmath.43.2006.1.5.

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Cyclic codes are defined as ideals in polynomial quotient rings. We are using a matrix ring construction in a similar way to define classes of codes. It is shown that all cyclic and all linear codes can be embedded as ideals in this construction. A formula for the largest Hamming weight of one-sided ideals in incidence rings is given. It is shown that every incidence ring defined by a directed graph always possesses a principal one-sided ideal that achieves the optimum Hamming weight.

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30

Basu, M., and S. Bagchi. "New Bounds on the Minimum Average Distance of Binary Codes." Journal of Scientific Research 2, no. 3 (2010): 489. http://dx.doi.org/10.3329/jsr.v2i3.2708.

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The minimum average Hamming distance of binary codes of length n and cardinality M is denoted by b(n,M). All the known lower bounds b(n,M) are useful when M is at least of size about 2n-1/n . In this paper, for large n, we improve upper and lower bounds for b(n,M). Keywords: Binary code; Hamming distance; Minimum average Hamming distance. © 2010 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. DOI: 10.3329/jsr.v2i3.2708 J. Sci. Res. 2 (3), 489-493 (2010)

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31

Olshevska, V. A. "Permutation codes over Sylow 2-subgroups $Syl_2(S_{2^n})$ of symmetric groups $S_{2^n}$." Researches in Mathematics 29, no. 2 (2021): 28. http://dx.doi.org/10.15421/242107.

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The permutation code (or the code) is well known object of research starting from 1970s. The code and its properties is used in different algorithmic domains such as error-correction, computer search, etc. It can be defined as follows: the set of permutations with the minimum distance between every pair of them. The considered distance can be different. In general, there are studied codes with Hamming, Ulam, Levensteins, etc. distances.In the paper we considered permutations codes over 2-Sylow subgroups of symmetric groups with Hamming distance over them. For this approach representation of permutations by rooted labeled binary trees is used. This representation was introduced in the previous author's paper. We also study the property of the Hamming distance defined on permutations from Sylow 2-subgroup $Syl_2(S_{2^n})$ of symmetric group $S_{2^n}$ and describe an algorithm for finding the Hamming distance over elements from Sylow 2-subgroup of the symmetric group with complexity $O(2^n)$. The metric properties of the codes that are defined on permutations from Sylow 2-subgroup $Syl_2(S_{2^n})$ of symmetric group $S_{2^n}$ are studied. The capacity and number of codes for the maximum and the minimum non-trivial distance over codes are characterized.

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32

Park, Sung Ik, and Kyeongcheol Yang. "Extended Hamming accumulate codes and modified irregular repeat accumulate codes." Electronics Letters 38, no. 10 (2002): 467. http://dx.doi.org/10.1049/el:20020316.

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33

van der Geer, G., and M. van der Vlugt. "Generalized Hamming Weights of Melas Codes and Dual Melas Codes." SIAM Journal on Discrete Mathematics 7, no. 4 (1994): 554–59. http://dx.doi.org/10.1137/s0895480193243365.

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34

MUNEMASA, Akihiro. "Homogeneity Theorems on Perfect Codes in Hamming Schemes and Generalized Hamming Schemes." Tokyo Journal of Mathematics 08, no. 2 (1985): 429–38. http://dx.doi.org/10.3836/tjm/1270151223.

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35

Manikandan, Mr G., and Dr M. Anand. "SEC-TAED based Error Detection and Correction Technique for Data Transmission Systems." Indonesian Journal of Electrical Engineering and Computer Science 10, no. 2 (2018): 696. http://dx.doi.org/10.11591/ijeecs.v10.i2.pp696-703.

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<p>In the OFDM communication system channel encoder and decoder is the part of the architecture. OFDM channel is mostly affected by Additive White Gaussian Noise (AWGN) in which bit flipping of original information leads to fault transmission in the channel. To overcome this problem by using hamming code for error detection and correction. Hamming codes are more attractive and it easy to process the encoding and decoding with low latency. In general the hamming is perfectly detected and corrects the single bit error. In this paper, design of single Error Correction-Triple Adjacent Error Detection (SEC-TAED) codes with bit placement algorithm is presented with less number of parity bits. In the conventional Double Adjacent Error Detection (DAED) and Hamming (13, 8) SEC-TAED are process the codes and detects the error, but it require more parity bits for performing the operation. The higher number of parity bits causes processing delay. To avoid this problem by proposed the Hamming (12, 8) SEC-TAED code, it require only four parity bits to perform the detection process. Bit-reordered format used in the method increases the probability detection of triple adjacent error. It is more suitable for efficient and high speed communication.</p>

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36

Gregor, Petr. "Hypercube 1-factorizations from extended Hamming codes." Electronic Notes in Discrete Mathematics 34 (August 2009): 627–31. http://dx.doi.org/10.1016/j.endm.2009.07.106.

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37

FAN, Yun. "Generalized Hamming weights and equivalences of codes." Science in China Series A 46, no. 5 (2003): 690. http://dx.doi.org/10.1360/02ys0368.

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38

van der Geer, G., and M. van der Vlugt. "On generalized Hamming weights of BCH codes." IEEE Transactions on Information Theory 40, no. 2 (1994): 543–46. http://dx.doi.org/10.1109/18.312183.

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39

Yang, Minghui, Jin Li, Keqin Feng, and Dongdai Lin. "Generalized Hamming Weights of Irreducible Cyclic Codes." IEEE Transactions on Information Theory 61, no. 9 (2015): 4905–13. http://dx.doi.org/10.1109/tit.2015.2444013.

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40

Johnsen, Trygve, and Hugues Verdure. "Generalized Hamming Weights for Almost Affine Codes." IEEE Transactions on Information Theory 63, no. 4 (2017): 1941–53. http://dx.doi.org/10.1109/tit.2017.2654456.

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41

Aiello, William, and F. T. Leighton. "Hamming Codes, Hypercube Embeddings, and Fault Tolerance." SIAM Journal on Computing 37, no. 3 (2007): 783–803. http://dx.doi.org/10.1137/s0097539798332464.

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42

Honkala, Iiro, and Antoine Lobstein. "On Identifying Codes in Binary Hamming Spaces." Journal of Combinatorial Theory, Series A 99, no. 2 (2002): 232–43. http://dx.doi.org/10.1006/jcta.2002.3263.

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43

Honary, B., and G. Markarian. "Low-complexity trellis decoding of Hamming codes." Electronics Letters 29, no. 12 (1993): 1114. http://dx.doi.org/10.1049/el:19930743.

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44

Zhang, Jun, and Keqin Feng. "Relative generalized Hamming weights of cyclic codes." Finite Fields and Their Applications 50 (March 2018): 338–55. http://dx.doi.org/10.1016/j.ffa.2017.12.008.

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45

Beelen, Peter, and Mrinmoy Datta. "Generalized Hamming weights of affine Cartesian codes." Finite Fields and Their Applications 51 (May 2018): 130–45. http://dx.doi.org/10.1016/j.ffa.2018.01.006.

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46

Gillespie, Neil I., and Cheryl E. Praeger. "Neighbour transitivity on codes in Hamming graphs." Designs, Codes and Cryptography 67, no. 3 (2012): 385–93. http://dx.doi.org/10.1007/s10623-012-9614-5.

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47

González Sarabia, Manuel, and Carlos Rentería Márquez. "Generalized Hamming weights and some parameterized codes." Discrete Mathematics 339, no. 2 (2016): 813–21. http://dx.doi.org/10.1016/j.disc.2015.10.026.

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48

Shutao, Xia, and Fu Fangwei. "Error detection capability of shortened hamming codes and their dual codes." Acta Mathematicae Applicatae Sinica 16, no. 3 (2000): 292–98. http://dx.doi.org/10.1007/bf02679894.

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49

Kéri, Gerzson. "On small covering codes in arbitrary mixed hamming spaces." Studia Scientiarum Mathematicarum Hungarica 44, no. 4 (2007): 517–34. http://dx.doi.org/10.1556/sscmath.2007.1027.

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Some results of Kéri and Östergård [3] regarding mixed binary/ternary covering codes are extended a) for mixed binary/ternary/quaternary codes, b) for codes in arbitrary mixed Hamming spaces. Some other results (sufficient conditions and examples of solved and unsolved cases) regarding 5-word codes in Z2bZ3tZ4q and Z2bZ3tZ4qZ5f are also provided in the paper.

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50

Wu, Yansheng, and Yoonjin Lee. "Self-Orthogonal Codes Constructed from Posets and Their Applications in Quantum Communication." Mathematics 8, no. 9 (2020): 1495. http://dx.doi.org/10.3390/math8091495.

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It is an important issue to search for self-orthogonal codes for construction of quantum codes by CSS construction (Calderbank-Sho-Steane codes); in quantum error correction, CSS codes are a special type of stabilizer codes constructed from classical codes with some special properties, and the CSS construction of quantum codes is a well-known construction. First, we employ hierarchical posets with two levels for construction of binary linear codes. Second, we find some necessary and sufficient conditions for these linear codes constructed using posets to be self-orthogonal, and we use these self-orthogonal codes for obtaining binary quantum codes. Finally, we obtain four infinite families of binary quantum codes for which the minimum distances are three or four by CSS construction, which include binary quantum Hamming codes with length n≥7. We also find some (almost) “optimal” quantum codes according to the current database of Grassl. Furthermore, we explicitly determine the weight distributions of these linear codes constructed using posets, and we present two infinite families of some optimal binary linear codes with respect to the Griesmer bound and a class of binary Hamming codes.

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