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Hamming (7,4) In coding theory, Hamming (7,4) is a linear error-correcting code that encodes four bits of data into seven bits by adding three parity bits. It is a member of a larger family of Hamming codes, but the term Hamming code often refers to this specific code that Richard W. Hamming introduced in 1950.
In mathematical terms, Hamming codes are a class of binary linear code. For each integer r ≥ 2 there is a code-word with block length n = 2r − 1 and message length k = 2r − r − 1. Hence the rate of Hamming codes is R = k / n = 1 − r / (2r − 1), which is the highest possible for codes with minimum distance of three (i.e., the minimal ...
Proprietary software for viewing and editing PDF documents. pdftk: GNU GPL/Proprietary: command-line tools to manipulate, edit and convert documents; supports filling of PDF forms with FDF/XFDF data. PDF-XChange Viewer: Freeware: Freeware PDF reader, tagger, editor (simple editions) and converter (free for non-commercial uses).
A generator matrix for a linear [,,]-code has format , where n is the length of a codeword, k is the number of information bits (the dimension of C as a vector subspace), d is the minimum distance of the code, and q is size of the finite field, that is, the number of symbols in the alphabet (thus, q = 2 indicates a binary code, etc.).
A convolutional code that is terminated is also a 'block code' in that it encodes a block of input data, but the block size of a convolutional code is generally arbitrary, while block codes have a fixed size dictated by their algebraic characteristics. Types of termination for convolutional codes include "tail-biting" and "bit-flushing".
Axiom. Axiom is a general-purpose computer algebra system. It has been in development since 1971 by IBM, and was originally named scratchpad. Richard Jenks originally headed it but over the years Barry Trager who then shaped the direction of the scratchpad project took over the project. It was eventually sold to the Numerical Algorithms Group ...
PDFtk (short for PDF Toolkit) is a toolkit for manipulating Portable Document Format (PDF) documents. [ 3][ 4] It runs on Linux, Windows and macOS. [ 5] It comes in three versions: PDFtk Server ( open-source command-line tool ), PDFtk Free ( freeware) and PDFtk Pro ( proprietary paid ). [ 2] It is able to concatenate, shuffle, split and rotate ...
The first two of its ten chapters present background and introductory material, including Hamming distance, decoding methods including maximum likelihood and syndromes, sphere packing and the Hamming bound, the Singleton bound, and the Gilbert–Varshamov bound, and the Hamming(7,4) code.