In coding theory, the weight enumerator polynomial of a binary linear code specifies the number of words of each possible Hamming weight.
Let be a binary linear code length . The weight distribution is the sequence of numbers
giving the number of codewords c in C having weight t as t ranges from 0 to n. The weight enumerator is the bivariate polynomial
Denote the dual code of by
(where denotes the vector dot product and which is taken over ).
The MacWilliams identity states that
The identity is named after Jessie MacWilliams.
The distance distribution or inner distribution of a code C of size M and length n is the sequence of numbers
where i ranges from 0 to n. The distance enumerator polynomial is
and when C is linear this is equal to the weight enumerator.
The outer distribution of C is the 2n-by-n+1 matrix B with rows indexed by elements of GF(2)n and columns indexed by integers 0...n, and entries
The sum of the rows of B is M times the inner distribution vector (A0,...,An).
A code C is regular if the rows of B corresponding to the codewords of C are all equal.
In commutative ring theory, a branch of mathematics, the radical of an ideal is an ideal such that an element is in the radical if and only if some power of is in . A radical ideal is an ideal that is equal to its own radical. The radical of a primary ideal is a prime ideal.
In computability theory, traditionally called recursion theory, a set S of natural numbers is called recursively enumerable, computably enumerable, semidecidable, provable or Turing-recognizable if:
In control theory and signal processing, a linear, time-invariant system is said to be minimum-phase if the system and its inverse are causal and stable.
In real analysis, the projectively extended real line, is the extension of the number line by a point denoted ∞. It is thus the set with the standard arithmetic operations extended where possible, sometimes denoted by The added point is called the point at infinity, because it is considered as a neighbour of both ends of the real line. More precisely, the point at infinity is the limit of every sequence of real numbers whose absolute values are increasing and unbounded.
In abstract algebra, the biquaternions are the numbers w + xi + yj + zk, where w, x, y, and z are complex numbers, or variants thereof, and the elements of {1, i, j, k} multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions corresponding to complex numbers and the variations thereof:
In coding theory, the dual code of a linear code
In coding theory, a linear code is an error-correcting code for which any linear combination of codewords is also a codeword. Linear codes are traditionally partitioned into block codes and convolutional codes, although turbo codes can be seen as a hybrid of these two types. Linear codes allow for more efficient encoding and decoding algorithms than other codes.
In coding theory, the Singleton bound, named after Richard Collom Singleton, is a relatively crude upper bound on the size of an arbitrary block code with block length , size and minimum distance .
In coding theory, decoding is the process of translating received messages into codewords of a given code. There have been many common methods of mapping messages to codewords. These are often used to recover messages sent over a noisy channel, such as a binary symmetric channel.
Reed–Muller codes are error-correcting codes that are used in wireless communications applications, particularly in deep-space communication. Moreover, the proposed 5G standard relies on the closely related polar codes for error correction in the control channel. Due to their favorable theoretical and mathematical properties, Reed–Muller codes have also been extensively studied in theoretical computer science.
A Z-channel is a communications channel used in coding theory and information theory to model the behaviour of some data storage systems.
In domain theory, a branch of mathematics and computer science, a Scott information system is a primitive kind of logical deductive system often used as an alternative way of presenting Scott domains.
In polynomial interpolation of two variables, the Padua points are the first known example of a unisolvent point set with minimal growth of their Lebesgue constant, proven to be O(log2 n) . Their name is due to the University of Padua, where they were originally discovered.
A locally decodable code (LDC) is an error-correcting code that allows a single bit of the original message to be decoded with high probability by only examining a small number of bits of a possibly corrupted codeword. This property could be useful, say, in a context where information is being transmitted over a noisy channel, and only a small subset of the data is required at a particular time and there is no need to decode the entire message at once. Note that locally decodable codes are not a subset of locally testable codes, though there is some overlap between the two.
DNA code construction refers to the application of coding theory to the design of nucleic acid systems for the field of DNA–based computation.
In discrete mathematics, ideal lattices are a special class of lattices and a generalization of cyclic lattices. Ideal lattices naturally occur in many parts of number theory, but also in other areas. In particular, they have a significant place in cryptography. Micciancio defined a generalization of cyclic lattices as ideal lattices. They can be used in cryptosystems to decrease by a square root the number of parameters necessary to describe a lattice, making them more efficient. Ideal lattices are a new concept, but similar lattice classes have been used for a long time. For example cyclic lattices, a special case of ideal lattices, are used in NTRUEncrypt and NTRUSign.
AN codes are error-correcting code that are used in arithmetic applications. Arithmetic codes were commonly used in computer processors to ensure the accuracy of its arithmetic operations when electronics were more unreliable. Arithmetic codes help the processor to detect when an error is made and correct it. Without these codes, processors would be unreliable since any errors would go undetected. AN codes are arithmetic codes that are named for the integers and that are used to encode and decode the codewords.
In coding theory, the Wozencraft ensemble is a set of linear codes in which most of codes satisfy the Gilbert-Varshamov bound. It is named after John Wozencraft, who proved its existence. The ensemble is described by Massey (1963), who attributes it to Wozencraft. Justesen (1972) used the Wozencraft ensemble as the inner codes in his construction of strongly explicit asymptotically good code.
In statistical learning theory, a representer theorem is any of several related results stating that a minimizer of a regularized empirical risk functional defined over a reproducing kernel Hilbert space can be represented as a finite linear combination of kernel products evaluated on the input points in the training set data.
In combinatorics, a lattice path in of length with steps in is a sequence such that each consecutive difference lies in . A lattice path may lie in any lattice in , but the integer lattice is most commonly used.
