New quantum codes from dual-containing cyclic codes over finite rings

Let \(R=\mathbb {F}_{2^{m}}+u\mathbb {F}_{2^{m}}+\cdots +u^{k}\mathbb {F}_{2^{m}}\), where \(\mathbb {F}_{2^{m}}\) is the finite field with \(2^{m}\) elements, m is a positive integer, and u is an indeterminate with \(u^{k+1}=0.\) In this paper, we propose the constructions of two new families of quantum codes obtained from dual-containing cyclic codes of odd length over R. A new Gray map over R is defined, and a sufficient and necessary condition for the existence of dual-containing cyclic codes over R is given. A new family of \(2^{m}\)-ary quantum codes is obtained via the Gray map and the Calderbank–Shor–Steane construction from dual-containing cyclic codes over R. In particular, a new family of binary quantum codes is obtained via the Gray map, the trace map and the Calderbank–Shor–Steane construction from dual-containing cyclic codes over R.     

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...     

Quantum error-correcting codes from algebraic geometry codes of Castle type

We study algebraic geometry codes producing quantum error-correcting codes by the CSS construction. We pay particular attention to the family of Castle codes. We show that many of the examples known in the literature in fact belong to this family of codes. We systematize these constructions by showing the common theory that underlies all of them.     

Conflict-avoiding codes and cyclic triple systems

The paper deals with the problem of constructing a code of the maximum possible cardinality consisting of binary vectors of length n and Hamming weight 3 and having the following property: any 3 × n matrix whose rows are cyclic shifts of three different code vectors contains a 3 × 3 permutation matrix as a submatrix. This property (in the special case w = 3) characterizes conflict-avoiding codes of length n for w active users, introduced in [1]. Using such codes in channels with asynchronous multiple access allows each of w active users to transmit a data packet successfully in one of w attempts during n time slots without collisions with other active users. An upper bound on the maximum cardinality of a conflict-avoiding code of length n with w = 3 is proved, and constructions of optimal codes achieving this bound are given. In particular, there are found conflict-avoiding codes for w = 3 which have much more vectors than codes of the same length obtained from cyclic Steiner triple systems by choosing a representative in each cyclic class.     

A new subclass of cyclic Goppa codes

We propose a subclass of cyclic Goppa codes given by separable self-reciprocal Goppa polynomials of degree two. We prove that this subclass contains all reversible cyclic codes of length n, n | (q ^m ± 1), with a generator polynomial g(x), g(α ^±i ) = 0, i = 0, 1, α ⁿ = 1, α ∈ GF(q ^2m ).     

A note on constant weight cyclic codes

In this paper we show that the dual code of a g-ary Hamming single error-correcting code is a constant weight cyclic code with weight q(m-1). We give, as a consequence, a very simple alternative derivation of the weight enumerator formula for the q-ary Hamming single error-correcting code. We show also that a binary cyclic equidistant code can be derived from this dual code     

Generalized error-locating codes with component codes over the same alphabet

We consider generalized error locating (GEL) codes over the same alphabet for both component codes. We propose an algorithm for computing an upper bound on the decoding error probability under known input symbol error rate and code parameters. Is is used to construct an algorithm for selecting code parameters to maximize the code rate for a given construction and given input and output error probabilities. A lower bound on the decoding error probability is given. Examples of plots of decoding error probability versus input symbol error rate are given, and their behavior is explained.     

Some properties of binaryn-Tuples with reference to cyclic codes

This article gives a geometric interpretation of a product (already introduced in literature) betweenn-tuples, based on the use of «associate, polynomials». In the second part of the work, a number of algebraic considerations are made, which give a characterization of the ringsR _n of all binaryn-tuples.     

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.     

噪声信道下的盲量子计算

盲量子计算(Blind Quantum Computation,BQC)区别于传统的量子计算(Quantum Metrology),它将客户端的计算任务通过量子信道委托给服务器端完成,解放客户端的计算压力,这就要求在信道的传输过程中,量子尽量精确传输.由于量子信道的噪声问题,理想情况下的无噪传输协议是不可能实现的,需要...     

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…     

u-Constacyclic codes over $${\mathbb {F}}_p+ u{\mathbb {F}}_p$$Fp+ uFp and their applications of constr ucting new non-binary q uant um codes

Structural properties of u-constacyclic codes over the ring \({\mathbb {F}}_p+u{\mathbb {F}}_p\) are given, where p is an odd prime and \(u^2=1\). Under a special Gray map from \({\mathbb {F}}_p+u{\mathbb {F}}_p\) to \({\mathbb {F}}_p^2\), some new non-binary quantum codes are obtained by this class of constacyclic codes.     

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.     

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.     

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.     

Decoding interleaved Reed–Solomon codes over noisy channels

We consider error correction over the Non-Binary Symmetric Channel (NBSC) which is a natural probabilistic extension of the Binary Symmetric Channel (BSC). We propose a new decoding algorithm for interleaved Reed–Solomon codes that attempts to correct all “interleaved” codewords simultaneously. In particular, interleaved encoding gives rise to multi-dimensional curves and more specifically to a variation of the Polynomial Reconstruction Problem, which we call Simultaneous Polynomial Reconstruction. We present and analyze a novel probabilistic algorithm that solves this problem. Our construction yields a decoding algorithm for interleaved RS codes that allows efficient transmission arbitrarily close to the channel capacity in the NBSC model.     

On the decoding of binary cyclic codes with the Newton identities

We revisit in this paper the concept of decoding binary cyclic codes with Gröbner bases. These ideas were first introduced by Cooper, then Chen, Reed, Helleseth and Truong, and eventually by Orsini and Sala. We discuss here another way of putting the decoding problem into equations: the Newton identities. Although these identities have been extensively used for decoding, the work was done manually, to provide formulas for the coefficients of the locator polynomial. This was achieved by Reed, Chen, Truong and others in a long series of papers, for decoding quadratic residue codes, on a case-by-case basis. It is tempting to automate these computations, using elimination theory and Gröbner bases.Thus, we study in this paper the properties of the system defined by the Newton identities, for decoding binary cyclic codes. This is done in two steps, first we prove some facts about the variety associated with this system, then we prove that the ideal itself contains relevant equations for decoding, which lead to formulas.Then we consider the so-called online Gröbner basis decoding, where the work of computing a Gröbner basis is done for each received word. It is much more efficient for practical purposes than preprocessing and substituting into the formulas. Finally, we conclude with some computational results, for codes of interesting length (about one hundred).