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When (mod ) an extended quadratic residue code is self-dual; otherwise it is equivalent but not equal to its dual. By the Gleason–Prange theorem (named for Andrew Gleason and Eugene Prange ), the automorphism group of an extended quadratic residue code has a subgroup which is isomorphic to either P S L 2 ( p ) {\displaystyle PSL_{2}(p)} or S ...
The linear code is defined by its generator matrix, which we choose to be a random generator matrix; that is, a matrix of elements which are chosen independently and uniformly over the field . Recall that in a linear code , the distance equals the minimum weight of a nonzero codeword.
Lorentz force F on a charged particle (of charge q) in motion (instantaneous velocity v). The E field and B field vary in space and time. The force F acting on a particle of electric charge q with instantaneous velocity v , due to an external electric field E and magnetic field B , is given by ( SI definition of quantities [ 1 ] ): [ 12 ]
In field theory, a primitive element of a finite field GF(q) is a generator of the multiplicative group of the field. In other words, α ∈ GF(q) is called a primitive element if it is a primitive (q − 1) th root of unity in GF(q); this means that each non-zero element of GF(q) can be written as α i for some natural number i.
In a polynomial code over () with code length and generator polynomial () of degree , there will be exactly code words. Indeed, by definition, p ( x ) {\displaystyle p(x)} is a code word if and only if it is of the form p ( x ) = g ( x ) ⋅ q ( x ) {\displaystyle p(x)=g(x)\cdot q(x)} , where q ( x ) {\displaystyle q(x)} (the quotient ) is of ...
The Reed–Solomon code is actually a family of codes, where every code is characterised by three parameters: an alphabet size , a block length, and a message length, with <. The set of alphabet symbols is interpreted as the finite field F {\displaystyle F} of order q {\displaystyle q} , and thus, q {\displaystyle q} must be a prime power .
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