Linear codes from quadratic forms
Nettetlinear 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. NettetLinear codes are algebraic codes, typically over a finite field, where the (symbol-wise) sum of two codewords is always a codeword and the (symbol-wise) multiplication of a codeword by a field element is also a codeword. Linear codes that are also block codes are linear block codes. All three codes in Section III.D.7 are linear block codes.
Linear codes from quadratic forms
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Nettetbut it can be viewed as a quadratic form over Q, Z p for p =2 , Q 2, R or C since −1 2 lies in each of those rings. In fact it can be viewed as a quadratic form over Z/nZ for any odd n,as−2 is invertible mod n whenever n is odd. The subject of quadratic forms is vast and central to many parts of mathematics, such as linear Nettet1. 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.
Nettet8. mai 2024 · The expression QTDQ = H, where H is the Hessian matrix of second partials, tells us that twice your quadratic expression is 4(x + 5 4y − 1)2 − 9 4(y − 8 3)2 where the third row has a coefficient 0. This is because of the 0 diagonal element in D. NettetQuadratic forms in general encode the so-called "quadric surfaces" such as ellipses, hyperbolic paraboloids, and so on. The principal axis theorem, also known as the spectral theorem, is one of the most important theorems in linear algebra! It is what allows us to "transform" the quadratic forms your professor mentioned.
Nettetof linear codes constructed from general quadratic forms over Fq. Before introduc-ing them, we give some notations for the standard quadratic form Q(x¯) defined above. For any vector x¯ = (x1,x2,...,xm), denote x¯ = (x1,x2,...,xr) and x¯ = (xr+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 …
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.
NettetLinear codes are algebraic codes, typically over a finite field, where the (symbol-wise) sum of two codewords is always a codeword and the (symbol-wise) multiplication of a codeword by a field element is also a codeword. Linear codes that are also block codes are linear block codes. All three codes in Section III.D.7 are linear block codes. gujarat tiles factoryNettetUniversity of Michigan. Jun 2015 - Jul 20152 months. Ann Arbor, Michigan, United States. Studied Graph Theory and the relationships between distinct objects on campus. Described vertices ... gujarat tiles infoNettet24. mar. 2024 · A linear code over a finite field with q elements F_q is a linear subspace C subset F_q^n. The vectors forming the subspace are called codewords. When codewords are chosen such that the distance between them is maximized, the code is called error-correcting since slightly garbled vectors can be recovered by choosing the … bowen police station qldNettet22. 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. gujarat times classifiedNettet7. 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): … bowen post office phone numberNettet1. mar. 2024 · Converting from quadratic form to standard form is quite common, so you can also check out this helpful video for another example. Return to the Table of Contents. Convert from Factored Form to Standard Form. To convert an equation from factored form into standard form simply involves multiplying the factors. bowenpress.comNettet1. jan. 2024 · Linear codes with a few weights have important applications in secret sharing, authentication codes, data storage system, association schemes, and strongly regular graphs. gujarat times of india