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OptimalEntanglement-AssistedQuantum CodesofSmallLength

Ching–Yi Lai, Todd A. Brun, MacWilliams Identity

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

• Consider the Galois Field GF (4) = {0, 1, ω, ω2 = ω} with 1 + ω + ω2 = 0 and ω3 = 1. • For g1 = ωu1 + ωv1, g2 = ωu2 + ωv2 ∈ GF (4)n, we define a symplectic inner product ∗ in GF (4)n by g1 ∗ g2 = u1 · v2 + u2 · v1 mod 2, where · is the usual binary inner product. • The orthogonal group of a subgroup C of GF (4)n with respect to ∗ is defined by C⊥ = {g ∈ GF (4) : g ∗ h = 0, ∀h ∈ C}. If C is an (n, 2n−k) additive code, then C⊥ is an (n, 2n+k) additive code.

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• Consider the Galois Field GF (4) = {0, 1, ω, ω2 = ω} with 1 + ω + ω2 = 0 and ω3 = 1. • For g1 = ωu1 + ωv1, g2 = ωu2 + ωv2 ∈ GF (4)n, we define a symplectic inner product ∗ in GF (4)n by g1 ∗ g2 = u1 · v2 + u2 · v1 mod 2, where · is the usual binary inner product. • The orthogonal group of a subgroup C of GF (4)n with respect to ∗ is defined by C⊥ = {g ∈ GF (4) : g ∗ h = 0, ∀h ∈ C}. If C is an (n, 2n−k) additive code, then C⊥ is an (n, 2n+k) additive code.
. In this paper we discuss the status of the classication of the extremal additive codes over GF (4) which are self-dual under the trace inner product. These codes are of interest because of their connection to quantum codes. There are 5 inequivalent codes of length 8, 8 of length 9, at least 56 of length 10, and exactly 1 of length 11. We discuss the methods of classication, the concepts of lengthening and shortening, and connections to binary codes. 1. Introduction An additive code C over GF (4) of length n is an additive subgroup of GF (4) n . As C is a free GF(2)-module, it has size 2 k for some 0 k 2n. We call C an (n; 2 k ) code. It has a basis, as a GF(2)-module, consisting of k basis vectors; a generator matrix of C will be a k n matrix with entries in GF (4) whose rows are a basis of C. Interest in additive codes over GF (4) has arisen because of their correspondence to quantum codes as described in [3]. There is a natural inner product arising from the trace m...
Additive code C over GF(4) of length n is an additive subgroup of GF(4)n. It is well known [4] that the problem of finding stabilizer quantum error-correcting codes is transformed into problem of finding additive self-orthogonal codes over the Galois field GF(4) under a trace inner product. Our purpose is to construct good additive self-dual codes of length 13 ≤ n ≤ 21. In this paper we classify all extremal (optimal) codes of lengths 13 and 14, and we construct many extremal codes of lengths 15 and 16. Also, we construct some new extremal codes of lengths 17,18,19, and 21. We give the current status of known extremal (optimal) additive self-dual codes of lengths 13 to 21.
The problem of finding stabilizer quantum-error-correcting codes can be transformed into the problem of finding classical self-orthogonal linear codes over the Galois field GF(4) under a Hermitian inner product.A classical cyclic code over the field GF(4) is self-orthogonal with respect to the Hermitian inner product iff the generator polynomial of its dual code is a factor of its generator polynomial.Based on this connection,a class of quantum cyclic codes is constructed by finding corresponding classical cyclic codes.Some examples are also given in detail.
In this paper, we propose quantum stabilizer codes based on a new construction of self-orthogonal trace-inner product codes over the Galois field with 4 elements (GF(4)). First, from any two binary vectors, we construct a generator matrix of linear codes whose components are over GF(4). We prove that the proposed linear codes comply with the self-orthogonal, trace-inner product. Then, we propose mapping tables to construct new quantum stabilizer codes by using linear codes. Comparison results show that our proposed quantum codes have various dimensions for any code length with the capacity for better errors correction relative to the referenced quantum codes.