Linear codes from quadratic forms

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August undefined, 2024

NettetThis paper presents a class of linear codes over finite fields $$F_q$$Fq with quadratic forms via a general construction and determines the explicit complete weight enumerators of these linear codes. Linear codes have been an interesting topic in both theory and practice for many years. NettetIn mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial).For example, + is a quadratic form in the variables x and y.The coefficients usually belong to a fixed field K, such as the real or complex numbers, and one speaks of a quadratic form over K.If =, and the quadratic …

linear algebra - Intuition/meaning behind quadratic forms

http://www-math.ucdenver.edu/~wcherowi/courses/m7823/linearcodes.pdf Nettet22. jul. 2024 · Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site palettenservice wismar

Several classes of linear codes and their weight distributions

Nettet12. aug. 2024 · A characteristic feature of the geometry of quadratic forms is the systematic use of the $ n ( n + 1 ) / 2 $-dimensional coefficient (parameter) space, in which the lattice $ \Lambda $ is represented by a point. Let. be a quadratic form with real coefficients $ a _ {ij} = a _ {ji} $ ($ i , j = 1, \dots, n $). Nettettors of linear codes constructed from general quadratic forms over Fq. Before in-troducing them, we give some notation for the standard quadratic form Q(¯x) de-noted above. For any vector ¯x = (x1,x2,...,xm), denote x¯′ = (x1,x2,...,xr) and x¯′′ = (x r+1,xr+2,...,xm), where r is the rank of Q(¯x). Thus Q(¯x) = Q(x¯′). Let Qˆ(¯x) = Nettet4. nov. 2015 · Linear codes with few weights have applications in secrete sharing, authentication codes, association schemes, and strongly regular graphs. In this paper, several classes of p-ary linear codes with two or three weights are constructed from quadratic Bent functions over the finite field $${\mathbb {F}}_p$$ , where p is an … palette mac eyeshadow

Weight hierarchies of a family of linear codes associated with ...

Linear codes with few weights from inhomogeneous quadratic functions ...

Nettet Nettet24. mar. 2024 · Any real quadratic form in variables may be reduced to the diagonal form (8) with by a suitable orthogonal point-transformation. Also, two real quadratic forms are equivalent under the group of linear transformations iff they have the same quadratic form rank and quadratic form signature . See also summit engineering cheyenne wyNettet1. jun. 2024 · In this paper, linear codes with few weights are constructed from inhomogeneous quadratic functions over the finite field \ ( {\mathrm {GF}} (p)\), where p is an odd prime. They include some... palettenservice reuther hasselroth

"Nettet7. des. 2024 · In this paper, we further determine the complete weight hierarchy of linear codes from a quadratic form over a finite field of even characteristic by carefully studying the behavior of the quadratic form on the subspaces of this field and its dual space, and complement the results of Li and Li. READ FULL TEXT Chao Liu 52 publications Dabin … " - Linear codes from quadratic forms

Linear codes from quadratic forms

Nettet8. feb. 2015 · 2 Answers Sorted by: 2 There's another approach: the Gauss decomposition consists to write the quadratic form on sum of squares. We have Q ( x, y) = 3 x 2 − ( 5 + c) x y + 2 c y 2 = 3 ( x − 5 + c 6 y) 2 + ( 2 c − ( 5 + c) 2 12) y 2 so the discussion is on the sign of α ( c) = 2 c − ( 5 + c) 2 12 if α ( c) > 0 then Q is positive definite Nettet22. des. 2015 · Quadratic forms appear in optimization problems of multi-variable functions. Also, the idea of positive definiteness and negative definiteness are based on the quadratic form concept. This gives rise to a test in order to distinguish the extreme points, namely Maximum, Minimum, and Saddle points.

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NettetQuadratic forms are a quiet element in the classification of Lie algebras, through Weyl chambers and so on. The short version is that any discussion involving Euclidean spaces that talks about reflections is using a quadratic form, typically the ordinary dot product. Nettet22. nov. 2024 · From the polynomial f over the quotient space, a non-degenerate quadratic form is induced. Some related results on the subspaces and quotient spaces are obtained. Based on these results, the...

NettetQuadratic forms 2 With respect to dual bases the matrix M∇ is the matrix of that linear transformation. The bilinear form is said to be nondegenerateif this transformation—or, equivalently, its matrix—is invertible. Any mapfrom f: V→b determinesatranspose map b bb b.Upon identifying withVbb, theform ∇ is symmetric if and only if the map is equal to its … Nettet3. apr. 2024 · In this paper, we study the vectorial bentness of an arbitrary quadratic form and construct two classes of linear codes of few weights from the quadratic forms. Let $q$ be a prime...

Nettet17. sep. 2024 · Quadratic forms. Given a matrix A of N demeaned data points, the symmetric covariance matrix C = 1 NAAT determines the variance in a particular direction. Vu = u ⋅ (Cu), where u is a unit vector defining the direction. More generally, a symmetric m × m matrix A defines a function q: Rm → R by. q(x) = x ⋅ (Ax). Nettet25. okt. 2024 · We have our quadratic form just in mat... Stack Exchange Network. Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, ... Linear Algebra - Quadratic Forms. Ask Question Asked 5 years, 5 months ago. Modified 5 years, ... Group set of commands as atomic transactions (C++) Should I ...

NettetLinear Codes In the V[n,q] setting, an important class of codes are the linear codes, these codes are the ones whose code words form a sub-vector space of V[n,q]. If the subspace of V[n,q] is k dimensional then we talk about the subspace as an [n,k]-code. (Note that the square brackets indicate a linear code).

Nettetq-ary linear codes with few weights employing general quadratic forms over the finite field Fq is proposed, where q is an odd prime power. This generalizes some earlier constructions of p-ary linear codes from quadratic bent functions over the prime field Fp,whereis anp odd prime. The complete weight enumerators of the resultant q-ary … summit engineering group spartanburg scIn this paper, several classes of two-weight or three-weight linear codes over $$ {{\\mathbb {F}}}_p$$ F p from quadratic or non-quadratic functions are constructed and their weight distributions are determined. From the constructed codes, we obtain some optimal linear codes with respect to the Singleton bound and the Griesmer bound. … paletten lampertheimNettet13. apr. 2024 · The emphasis is on complex codes that take weeks or months to run, and that have a large number of undetermined input parameters; many climate prediction models fall into this class. The emulator essentially determines Bayesian posterior estimates of the PDF of the output of a model, conditioned on results from previous … summit energy weymouth ma summit engineering fort mill scNettet1. mar. 2024 · We called η ( Δ f), denoted by ϵ f, the sign of the quadratic form f. For a subspace H ⊆ F p m, define H ⊥ = { x ∈ F p m: F ( x, y) = 0 for each y ∈ H }. Then H ⊥ is called the dual space of H. And R f can also be defined as the codimension of F p m ⊥. Namely, R f + dim ( F p m ⊥) = m. Let H be a d -dimensional subspace of F p m. palettenshop24Nettet7. des. 2024 · The generalized Hamming weight of linear codes is a natural generalization of the minimum Hamming distance. They convey the structural information of a linear code and determine its performance in various applications, and have become one of important research topics in coding theory. Recently, Li (IEEE Trans. Inf. Theory, 67(1): … summit engineering south carolinaNettetlinear codes from a quadratic form overa ﬁnite ﬁeld ofeven characteristicby carefully studying the behavior of the quadratic form on the subspaces of this ﬁeld and its dual space, and complement the results of Li and Li. Keywords. Generalized Hamming weight, weight hierarchy, linear code, quadratic form. paletten schroth crailsheim