Self-Orthogonality of $q$-Ary Images of $q^{m}$-Ary Codes and Quantum Code Construction

Abstract

A code over GF(q m ) can be imaged or expanded into a code over GF(q) using a basis for the extension field over the base field. The properties of such an image depend on the original code and the basis chosen for imaging.Problems relating the properties of a code and its image with respect to a basis have been of great interest in the field of coding theory. In this work, a generalized version of the problem of self-orthogonality of the q-ary image of a q m -ary code has been considered. Given an inner product (more generally, a biadditive form), necessary and sufficient conditions have been derived for a code over a field extension and an expansion basis so that an image of that code is self-orthogonal. The conditions require that the original code be self-orthogonal with respect to several related biadditive forms whenever certain power sums of the dual basis elements do not vanish. Numerous interesting corollaries have been derived by specializing the general conditions. An interesting result for the canonical or regular inner product in fields of characteristic two is that only self-orthogonal codes result in self-orthogonal images. Another result is that image of a code is self-orthogonal for all bases if and only if trace of the code is self-orthogonal, except for the case of binary images of 4-ary codes. The conditions are particularly simple to state and apply for cyclic codes. To illustrate a possible application, new quantum error-correcting codes have been constructed with larger minimum distance than previously known.

Quantum Codes Derived from Two Construction Methods

“…FROM TWO METHODS From Theorems 2.1 and 2.3, we conclude that quantum code can be constructed from both codes over F q with Euclidean dual containing and codes over F q 2 with hermitian dual containing.The former, the CSS construction was first introduced in 1996 in [5][6], for a long time it is preferred for the elegant and relatively simpler form.On the other hand, the latter one requires extension field F q 2 , it seems more complex.But recently, hermitianorthogonality has received some attention [21]- [23].However considering the constructing conditions and the constructed quantum codes, how do the two methods compare remains unknown.Now we will compare the two methods in the same kind of quantum code.…”

Metodos de construção de codigos quanticos CSS e conexões entre codigos quanticos e matroides

“…Essas famílias generalizam as construções de códigos CSS binários apresentadas em [3] e possuem taxa de codificação tendendo ao valor 1. Além disso, na Tabela 2.2, exibimos alguns novos códigos quânticos construídos pelo métodos propostos neste capítulo, que estendem uma família de códigos apresentados em [19].…”

“…Os códigos cíclicos são exaustivamente estudados na literatura não somente no caso clássico bem como no caso quântico, veja, por exemplo, [4,6,7,13,16,19,20,29,30,[37][38][39][40][41][42][43][44].…”

“…Os códigos CSS formam uma subclasse contida na classe de códigos estabilizadores [1], e tal subclasse é a principal ferramenta utilizada para a construção de bons códigos quânticos, [2,[4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Poucos trabalhos existentes na literatura não utilizam a construção CSS para a geração de códigos quânticos, como, por exemplo, [21][22][23][24].…”

“…Em relação à criação de novos códigos quânticos, a maioria dos trabalhos apresentados na literatura são baseados em construções de códigos CSS derivados de um único código clássico auto-ortogonal C [2,[5][6][7][8][9][10][11][12][13][14][15][17][18][19][20]. Em [11] são apresentados códigos CSS derivados de um único código convolucional clássico, auto-ortogonal.…”