Construction of [[n,n − 4,3]]q quantum codes for odd prime power q
نویسندگان
چکیده
The theory of quantum error-correcting codes (QECCs, for short) has been exhaustively studied in the literature; see [1–8]. The most widely studied class of quantum codes are binary quantum stabilizer codes. A thorough discussion on the principles of quantum coding theory was given in [3] and [4] for binary quantum stabilizer codes. An appealing aspect of binary quantum codes is that there exist links to classical coding theory which make easy the construction of good quantum codes [8]. More recently similar theories of nonbinary quantum stabilizer codes were established in [6–8]; characterization of nonbinary quantum stabilizer codes over Fq (the finite field with q elements) in terms of classical codes over Fq2 was also given. Based on [6–8], many nonbinary quantum stabilizer codes were constructed from classical nonbinary codes; see [6–8] and references therein. One central theme in quantum error correction is the construction of quantum codes with good parameters [1–22]. Among these codes, quantum maximal-distance-separable (MDS) codes received much attention. Quantum MDS codes are optimal quantum codes, since they meet the quantum Singleton bound. Lemma 1.1 (quantum Singleton bound [5,8]). An [[n,k,d]]q quantum stabilizer code satisfies
منابع مشابه
On [[n,n-4,3]]q Quantum MDS Codes for odd prime power q
For each odd prime power q, let 4 ≤ n ≤ q2 + 1. Hermitian self-orthogonal [n, 2, n − 1] codes over GF (q2) with dual distance three are constructed by using finite field theory. Hence, [[n,n − 4, 3]]q quantum MDS codes for 4 ≤ n ≤ q2 + 1 are obtained.
متن کاملA Construction of MDS Quantum Convolutional Codes
In this paper, two new families of MDS quantum convolutional codes are constructed. The first one can be regarded as a generalization of [36, Theorem 6.5], in the sense that we do not assume that q ≡ 1 (mod 4). More specifically, we obtain two classes of MDS quantum convolutional codes with parameters: (i) [(q + 1, q − 4i + 3, 1; 2, 2i + 2)]q , where q ≥ 5 is an odd prime power and 2 ≤ i ≤ (q −...
متن کاملThe Explicit Construction of Irreducible Polynomials over Finite Fields
For a finite field GF(q) of odd prime power order q, and n > 1, we construct explicitly a sequence of monic irreducible reciprocal polynomials o f degree n2 m (m = 1, 2, 3 . . . . ) over GF(q). It is the analog for fields of odd order of constructions of Wiedemann and of Meyn over GF(2). We also deduce iterated presenn2** tations of GF(q ).
متن کاملOn the linear complexity of Legendre-Sidelnikov sequences
Background • Legendre Sequence For a prime p > 2 let (s n) be the Legendre sequence defined as s n = 1, n p = −1, 0, otherwise, n ≥ 0, where. p denotes the Legendre symbol. • Sidelnikov Sequence Let q be an odd prime power, g a primitive element of F q , and let η denote the quadratic character of F Background • Legendre Sequence For a prime p > 2 let (s n) be the Legendre sequence defined as s...
متن کاملOn Silverman's conjecture for a family of elliptic curves
Let $E$ be an elliptic curve over $Bbb{Q}$ with the given Weierstrass equation $ y^2=x^3+ax+b$. If $D$ is a squarefree integer, then let $E^{(D)}$ denote the $D$-quadratic twist of $E$ that is given by $E^{(D)}: y^2=x^3+aD^2x+bD^3$. Let $E^{(D)}(Bbb{Q})$ be the group of $Bbb{Q}$-rational points of $E^{(D)}$. It is conjectured by J. Silverman that there are infinitely many primes $p$ for which $...
متن کامل