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Hamming graphs are a special class of graphs named after Richard Hamming and used in several branches of mathematics (graph theory) and computer science. Let S be a set of q elements and d a positive integer. The Hamming graph H(d,q) has vertex set S d, the set of ordered d-tuples of elements of S, or sequences of length d from S.
For the outer code , a Reed-Solomon code would have been the first code to have come in mind. However, we would see that the construction of such a code cannot be done in polynomial time. This is why a binary linear code is used for .
Convolutional code with any code rate can be designed based on polynomial selection; [15] however, in practice, a puncturing procedure is often used to achieve the required code rate. Puncturing is a technique used to make a m/n rate code from a "basic" low-rate (e.g., 1/n) code. It is achieved by deleting of some bits in the encoder output.
For practical purposes, parity-check matrix of a binary Goppa code is usually converted to a more computer-friendly binary form by a trace construction, that converts the -by-matrix over () to a -by-binary matrix by writing polynomial coefficients of () elements on successive rows.
The scikit-learn Python package implements some multi-labels algorithms and metrics. The scikit-multilearn Python package specifically caters to the multi-label classification. It provides multi-label implementation of several well-known techniques including SVM, kNN and many more .
A PDF page description can use a matrix to scale, rotate, or skew graphical elements. A key concept in PDF is that of the graphics state, which is a collection of graphical parameters that may be changed, saved, and restored by a page description. PDF has (as of version 2.0) 25 graphics state properties, of which some of the most important are:
Hamming limit [clarification needed] There are theoretical limits (such as the Hamming limit), but another question is which codes can actually constructed. [clarification needed] It is like packing spheres in a box in many dimensions. This diagram shows the constructible codes, which are linear and binary.
A typical example of linear code is the Hamming code. Codes defined via a Hamming space necessarily have the same length for every codeword, so they are called block codes when it is necessary to distinguish them from variable-length codes that are defined by unique factorization on a monoid.