Galois环上循环码新的表示

近年来,在编码理论中,Galois环上码的研究成为编码理论工作者研究的一个热点。定义了Galois环上循环码的离散傅里叶变换及Mattson-Solomon（MS）多项式,证明了Galois环上的循环码同构于Galois环的Galois 扩张的理想。     

Quantum synchronizable codes from finite rings

Quantum Information Processing - The novel quantum dialogue (QD) protocol by using the three-qubit GHZ states (Quantum Inf Process (2019) 18: 37) is cryptanalyzed. It is found that there is the...     

A class of 1-generator quasi-cyclic codes over finite chain rings

Using another method, which is different with the one given by Cao [1-generator quasi-cyclic (QC) codes over finite chain rings, Appl. Algebra Eng. Commun. Comput. 24 (2013), pp. 53–72], we investigate the structural properties of a class of 1-generator QC codes over finite chain rings. We give the structure of the annihilator of 1-generator QC codes, the conditions for 1-generator QC codes to be free and the minimum distance bounds on 1-generator QC codes. Under some conditions, we also discuss the enumeration of 1-generator QC codes and describe how to obtain the one and only one generator for each 1-generator QC code.     

New quantum codes constructed from quaternary BCH codes

In this paper, we firstly study construction of new quantum error-correcting codes (QECCs) from three classes of quaternary imprimitive BCH codes. As a result, the improved maximal designed distance of these narrow-sense imprimitive Hermitian dual-containing quaternary BCH codes are determined to be much larger than the result given according to Aly et al. (IEEE Trans Inf Theory 53:1183–1188, 2007) for each different code length. Thus, families of new QECCs are newly obtained, and the constructed QECCs have larger distance than those in the previous literature. Secondly, we apply a combinatorial construction to the imprimitive BCH codes with their corresponding primitive counterpart and construct many new linear quantum codes with good parameters, some of which have parameters exceeding the finite Gilbert–Varshamov bound for linear quantum codes.     

New asymmetric quantum codes over $$F_{q}$$

Two families of new asymmetric quantum codes are constructed in this paper. The first family is the asymmetric quantum codes with length \(n=q^{m}-1\) over \(F_{q}\), where \(q\ge 5\) is a prime power. The second one is the asymmetric quantum codes with length \(n=3^{m}-1\). These asymmetric quantum codes are derived from the CSS construction and pairs of nested BCH codes. Moreover, let the defining set \(T_{1}=T_{2}^{-q}\), then the real Z-distance of our asymmetric quantum codes are much larger than \(\delta _\mathrm{max}+1\), where \(\delta _\mathrm{max}\) is the maximal designed distance of dual-containing narrow-sense BCH code, and the parameters presented here have better than the ones available in the literature.     

Quantum codes from cyclic codes over $$F_q+uF_q+vF_q+uvF_q$$

In this paper, we study quantum codes over \(F_q\) from cyclic codes over \(F_q+uF_q+vF_q+uvF_q,\) where \(u^2=u,~v^2=v,~uv=vu,~q=p^m\), and p is an odd prime. We give the structure of cyclic codes over \(F_q+uF_q+vF_q+uvF_q\) and obtain self-orthogonal codes over \(F_q\) as Gray images of linear and cyclic codes over \(F_q+uF_q+vF_q+uvF_q\). In particular, we decompose a cyclic code over \(F_q+uF_q+vF_q+uvF_q\) into four cyclic codes over \(F_q\) to determine the parameters of the corresponding quantum code.     

Entanglement-assisted quantum MDS codes from negacyclic codes

The entanglement-assisted formalism generalizes the standard stabilizer formalism, which can transform arbitrary classical linear codes into entanglement-assisted quantum error-correcting codes (EAQECCs) by using pre-shared entanglement between the sender and the receiver. In this work, we construct six classes of q-ary entanglement-assisted quantum MDS (EAQMDS) codes based on classical negacyclic MDS codes by exploiting two or more pre-shared maximally entangled states. We show that two of these six classes q-ary EAQMDS have minimum distance more larger than \(q+1\). Most of these q-ary EAQMDS codes are new in the sense that their parameters are not covered by the codes available in the literature.     

Entanglement-assisted quantum MDS codes constructed from constacyclic codes

Recently, entanglement-assisted quantum error-correcting codes (EAQECCs) have been constructed by cyclic codes and negacyclic codes. In this paper, by decomposing the defining set of constacyclic codes, we construct four classes of new EAQECCs, which satisfy the entanglement-assisted quantum Singleton bound.     

Asymmetric quantum codes: new codes from old

In this paper we extend to asymmetric quantum error-correcting codes the construction methods, namely: puncturing, extending, expanding, direct sum and the $({ \mathbf u}| \mathbf{u}+{ \mathbf v})$ construction. By applying these methods, several families of asymmetric quantum codes can be constructed. Consequently, as an example of application of quantum code expansion developed here, new families of asymmetric quantum codes derived from generalized Reed-Muller codes, quadratic residue, Bose-Chaudhuri-Hocquenghem, character codes and affine-invariant codes are constructed.     

Entanglement-assisted quantum MDS codes constructed from negacyclic codes

Recently, entanglement-assisted quantum codes have been constructed from cyclic codes by some scholars. However, how to determine the number of shared pairs required to construct entanglement-assisted quantum codes is not an easy work. In this paper, we propose a decomposition of the defining set of negacyclic codes. Based on this method, four families of entanglement-assisted quantum codes constructed in this paper satisfy the entanglement-assisted quantum Singleton bound, where the minimum distance satisfies \(q+1 \le d\le \frac{n+2}{2}\). Furthermore, we construct two families of entanglement-assisted quantum codes with maximal entanglement.     

Construction of nonbinary quantum cyclic codes by using graph method

~~Construction of nonbinary quantum cyclic codes by using graph method1. Wootters, W. K., Zurek, W. H., A single quantum cannot be cloned, Nature, 1982, 299: 802-803. 2. Shor, P. W., Scheme for reducing decoherence in quantum memory, Phys. Rev. A, 1995, 52: 2493. 3. Steane, A. M., Multiple particle interference and quantum error correction, Proc. Roy. Soc. London A, 1996, 452: 2551-2557. 4. Calderbank, A. R., Rains, E. M., Shor, P. W. et al., Quantum error correction via c…     

RaRb Transformation of Compound Finite Automata over Commutative Rings

Some results on RaRb transformation of compound finite automata over finite field are generalized to the case of commutative rings.Properties of RaRb transformation are discussed and applied to the inversion problem for compound finite automata.     

Constructing quantum codes

Quantum error correcting codes are indispensable for quantum information processing and quantum computation. In 1995 and 1996, Shor and Steane gave first several examples of quantum codes from classical error correcting codes. The construction of efficient quantum codes is now an active multi-discipline research field. In this paper we review the known several constructions of quantum codes and present some examples.     

Constructing quantum codes

Quantum error correcting codes are indispensable for quantum information processing and quantum computation. In 1995 and 1996, Shor and Steane gave first several examples of quantum codes from classical error correcting codes. The construction of efficient quantum codes is now an active multi-discipline research field. In this paper we review the known several constructions of quantum codes and present some examples.     

Some negacyclic BCH codes and quantum codes

Quantum Information Processing - Any quantum communication task requires a common reference frame (i.e., phase, coordinate system). In particular, quantum key distribution requires different bases...     

Construction of punctured and extended quantum codes over GF(2)

On the basis of elementary transformation,we propose a new method for constructing a class of pure quantum codes [[n-i,2k-n+i,d-i]]2 and [[n+1,2k-n-1,d+1]]2 from a class of classical linear codes [n,k,d]2 that contain their dual codes.The construction process was based on the elementary algebra;the error-correcting performance of the quantum codes was analyzed according to the relationship between the parity-check matrix and the minimum distance of the classical linear codes;the encoding and decoding networks were constructed based on the stabilizer.The proposed method is simple,straightforward and easy to implement using a computer and other hardware system.The theoretical results showed that the method is practical for the construction of a class of quantum codes.     

Equidistributed sequences over finite fields produced by one class of linear recurring sequences over residue rings

We consider the distribution of r-patterns in one class of uniformly distributed sequences over a finite field. We establish bounds for the number of occurrences of a given r-pattern and prove upper bounds for the cross-correlation function of these sequences.