A relation between the average Hamming distance and the average Hamming weight of binary codes

2001 ◽  
Vol 94 (2) ◽  
pp. 413-419 ◽  
Author(s):  
Zhao Zhi Zhang
Keyword(s):  
Hamming Distance ◽  
Binary Codes ◽  
Hamming Weight

scholarly journals New Bounds on the Minimum Average Distance of Binary Codes

2010 ◽  
Vol 2 (3) ◽  
pp. 489
Author(s):  
M. Basu ◽  
S. Bagchi
Keyword(s):  
Lower Bounds ◽  
Binary Code ◽  
Hamming Distance ◽  
Average Distance ◽  
Upper And Lower Bounds ◽  
Binary Codes ◽  
Large N

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) 

scholarly journals Deep Polarized Network for Supervised Learning of Accurate Binary Hashing Codes

2020 ◽  
Author(s):  
Lixin Fan ◽  
Kam Woh Ng ◽  
Ce Ju ◽  
Tianyu Zhang ◽  
Chee Seng Chan
Keyword(s):  
Large Deviations ◽  
Hamming Distance ◽  
State Of The Art ◽  
Metric Learning ◽  
Binary Codes ◽  
Distance Metric Learning ◽  
Learning Framework ◽  
Label Information ◽  
Learning To Hash ◽  
Random Codes

This paper proposes a novel deep polarized network (DPN) for learning to hash, in which each channel in the network outputs is pushed far away from zero by employing a differentiable bit-wise hinge-like loss which is dubbed as polarization loss. Reformulated within a generic Hamming Distance Metric Learning framework [Norouzi et al., 2012], the proposed polarization loss bypasses the requirement to prepare pairwise labels for (dis-)similar items and, yet, the proposed loss strictly bounds from above the pairwise Hamming Distance based losses. The intrinsic connection between pairwise and pointwise label information, as disclosed in this paper, brings about the following methodological improvements: (a) we may directly employ the proposed differentiable polarization loss with no large deviations incurred from the target Hamming distance based loss; and (b) the subtask of assigning binary codes becomes extremely simple --- even random codes assigned to each class suffice to result in state-of-the-art performances, as demonstrated in CIFAR10, NUS-WIDE and ImageNet100 datasets.

scholarly journals Discrete Social Recommendation

2019 ◽  
Vol 33 ◽  
pp. 208-215 ◽  
Author(s):  
Chenghao Liu ◽  
Xin Wang ◽  
Tao Lu ◽  
Wenwu Zhu ◽  
Jianling Sun ◽  
...  
Keyword(s):  
Hamming Distance ◽  
Social Information ◽  
Original Data ◽  
Binary Codes ◽  
Model Parameters ◽  
Social Recommendation ◽  
Unified Framework ◽  
Latent Features ◽  
Real Value ◽  
Real World Datasets

Social recommendation, which aims at improving the performance of traditional recommender systems by considering social information, has attracted broad range of interests. As one of the most widely used methods, matrix factorization typically uses continuous vectors to represent user/item latent features. However, the large volume of user/item latent features results in expensive storage and computation cost, particularly on terminal user devices where the computation resource to operate model is very limited. Thus when taking extra social information into account, precisely extracting K most relevant items for a given user from massive candidates tends to consume even more time and memory, which imposes formidable challenges for efficient and accurate recommendations. A promising way is to simply binarize the latent features (obtained in the training phase) and then compute the relevance score through Hamming distance. However, such a two-stage hashing based learning procedure is not capable of preserving the original data geometry in the real-value space and may result in a severe quantization loss. To address these issues, this work proposes a novel discrete social recommendation (DSR) method which learns binary codes in a unified framework for users and items, considering social information. We further put the balanced and uncorrelated constraints on the objective to ensure the learned binary codes can be informative yet compact, and finally develop an efficient optimization algorithm to estimate the model parameters. Extensive experiments on three real-world datasets demonstrate that DSR runs nearly 5 times faster and consumes only with 1/37 of its real-value competitor’s memory usage at the cost of almost no loss in accuracy.

scholarly journals On the average Hamming distance for binary codes

1998 ◽  
Vol 89 (1-3) ◽  
pp. 269-276 ◽  
Author(s):  
Xia Shutao ◽  
Fangwei Fu
Keyword(s):  
Hamming Distance ◽  
Binary Codes

scholarly journals Efficient Querying from Weighted Binary Codes

2020 ◽  
Vol 34 (07) ◽  
pp. 12346-12353
Author(s):  
Zhenyu Weng ◽  
Yuesheng Zhu
Keyword(s):  
Great Improvement ◽  
Large Scale ◽  
Hamming Distance ◽  
New Method ◽  
Binary Codes ◽  
Search Efficiency ◽  
Efficient Computation ◽  
Hash Tables ◽  
Multi Index ◽  
Open Issue

Binary codes are widely used to represent the data due to their small storage and efficient computation. However, there exists an ambiguity problem that lots of binary codes share the same Hamming distance to a query. To alleviate the ambiguity problem, weighted binary codes assign different weights to each bit of binary codes and compare the binary codes by the weighted Hamming distance. Till now, performing the querying from the weighted binary codes efficiently is still an open issue. In this paper, we propose a new method to rank the weighted binary codes and return the nearest weighted binary codes of the query efficiently. In our method, based on the multi-index hash tables, two algorithms, the table bucket finding algorithm and the table merging algorithm, are proposed to select the nearest weighted binary codes of the query in a non-exhaustive and accurate way. The proposed algorithms are justified by proving their theoretic properties. The experiments on three large-scale datasets validate both the search efficiency and the search accuracy of our method. Especially for the number of weighted binary codes up to one billion, our method shows a great improvement of more than 1000 times faster than the linear scan.

scholarly journals Calculating Hamming Distance with the IBM Q Experience

2018 ◽  
Author(s):  
José Manuel Bravo
Keyword(s):  
Open Access ◽  
Hamming Distance ◽  
Quantum Computer ◽  
Quantum Algorithm ◽  
Equal Length ◽  
Hamming Weight ◽  
Xor Operation ◽  
Binary Strings

In this brief paper a quantum algorithm to calculate the Hamming distance of two binary strings of equal length (or messages in theory information) is presented. The algorithm calculates the Hamming weight of two binary strings in one query of an oracle. To calculate the hamming distance of these two strings we only have to calculate the Hamming weight of the xor operation of both strings. To test the algorithms the quantum computer prototype that IBM has given open access to on the cloud has been used to test the results.

scholarly journals Top Position Sensitive Ordinal Relation Preserving Bitwise Weight for Image Retrieval

Algorithms ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 18
Author(s):  
Zhen Wang ◽  
Fuzhen Sun ◽  
Longbo Zhang ◽  
Lei Wang ◽  
Pingping Liu
Keyword(s):  
Large Scale ◽  
Hamming Distance ◽  
Superior Performance ◽  
Binary Codes ◽  
Training Samples ◽  
Benchmark Datasets ◽  
Data Points ◽  
Position Sensitive ◽  
Core Idea ◽  
Optimization Mechanism

In recent years, binary coding methods have become increasingly popular for tasks of searching approximate nearest neighbors (ANNs). High-dimensional data can be quantized into binary codes to give an efficient similarity approximation via a Hamming distance. However, most of existing schemes consider the importance of each binary bit as the same and treat training samples at different positions equally, which causes many data pairs to share the same Hamming distance and a larger retrieval loss at the top position. To handle these problems, we propose a novel method dubbed by the top-position-sensitive ordinal-relation-preserving bitwise weight (TORBW) method. The core idea is to penalize data points without preserving an ordinal relation at the top position of a ranking list more than those at the bottom and assign different weight values to their binary bits according to the distribution of query data. Specifically, we design an iterative optimization mechanism to simultaneously learn binary codes and bitwise weights, which makes their learning processes related to each other. When the iterative procedure converges, the binary codes and bitwise weights are effectively adapted to each other. To reduce the training complexity, we relax the discrete constraints of both the binary codes and the indicator function. Furthermore, we pretrain a tensor ordinal graph to decrease the time consumption of computing a relative similarity relationship among data points. Experimental results on three large-scale ANN search benchmark datasets, i.e., SIFT1M, GIST1M, and Cifar10, show that the proposed TORBW method can achieve superior performance over state-of-the-art approaches.

scholarly journals On a Class of Constant Weight Codes

10.37236/1228 ◽  
1996 ◽  
Vol 3 (1) ◽  
Author(s):  
Mihai Caragiu
Keyword(s):  
Prime Power ◽  
Binary Code ◽  
Hamming Distance ◽  
Character Sums ◽  
Bch Codes ◽  
Binary Codes ◽  
Quadratic Character ◽  
Constant Weight Codes ◽  
Constant Weight ◽  
Non Linear

For any odd prime power $q$ we first construct a certain non-linear binary code $C(q,2)$ having $(q^2-q)/2$ codewords of length $q$ and weight $(q-1)/2$ each, for which the Hamming distance between any two distinct codewords is in the range $[q/2-3\sqrt q/2,\ q/2+3\sqrt q/2]$ that is, 'almost constant'. Moreover, we prove that $C(q,2)$ is distance-invariant. Several variations and improvements on this theme are then pursued. Thus, we produce other classes of binary codes $C(q,n)$, $n\geq 3$, of length $q$ that have 'almost constant' weights and distances, and which, for fixed $n$ and big $q$, have asymptotically $q^n/n$ codewords. Then we prove the possibility of extending our codes by adding the complements of their codewords. Also, by using results on Artin $L-$series, it is shown that the distribution ofthe $0$'s and $1$'s in the codewords we constructed is quasi-random. Our construction uses character sums associated with the quadratic character $\chi$ of $F_{q^n}$ in which the range of summation is $F_q$. Relations with the duals of the double error correcting BCH codes and the duals of the Melas codes are also discussed.

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