Quantum Error Correction via Codes over GF(4)

The problem of finding quantum error-correcting codes is transformed into the problem of finding additive codes over the field GF(4) which are self-orthogonal with respect to a certain trace inner product. Many new codes and new bounds are presented, as well as a table of upper and lower bounds on such codes of length up to 30 qubits.Comment: Latex, 46 pages. To appear in IEEE Transactions on Information Theory. Replaced Sept. 24, 1996, to correct a number of minor errors. Replaced Sept. 10, 1997. The second section has been completely rewritten, and should hopefully be much clearer. We have also added a new section discussing the developments of the past year. Finally, we again corrected a number of minor error

Self-Dual Codes

Self-dual codes are important because many of the best codes known are of this type and they have a rich mathematical theory. Topics covered in this survey include codes over F_2, F_3, F_4, F_q, Z_4, Z_m, shadow codes, weight enumerators, Gleason-Pierce theorem, invariant theory, Gleason theorems, bounds, mass formulae, enumeration, extremal codes, open problems. There is a comprehensive bibliography.Comment: 136 page

The invariants of the Clifford groups

The automorphism group of the Barnes-Wall lattice L_m in dimension 2^m (m not 3) is a subgroup of index 2 in a certain ``Clifford group'' C_m (an extraspecial group of order 2^(1+2m) extended by an orthogonal group). This group and its complex analogue CC_m have arisen in recent years in connection with the construction of orthogonal spreads, Kerdock sets, packings in Grassmannian spaces, quantum codes, Siegel modular forms and spherical designs. In this paper we give a simpler proof of Runge's 1996 result that the space of invariants for C_m of degree 2k is spanned by the complete weight enumerators of the codes obtained by tensoring binary self-dual codes of length 2k with the field GF(2^m); these are a basis if m >= k-1. We also give new constructions for L_m and C_m: let M be the Z[sqrt(2)]-lattice with Gram matrix [2, sqrt(2); sqrt(2), 2]. Then L_m is the rational part of the mth tensor power of M, and C_m is the automorphism group of this tensor power. Also, if C is a binary self-dual code not generated by vectors of weight 2, then C_m is precisely the automorphism group of the complete weight enumerator of the tensor product of C and GF(2^m). There are analogues of all these results for the complex group CC_m, with ``doubly-even self-dual code'' instead of ``self-dual code''.Comment: Latex, 24 pages. Many small improvement

Convolutional and tail-biting quantum error-correcting codes

Rate-(n-2)/n unrestricted and CSS-type quantum convolutional codes with up to 4096 states and minimum distances up to 10 are constructed as stabilizer codes from classical self-orthogonal rate-1/n F_4-linear and binary linear convolutional codes, respectively. These codes generally have higher rate and less decoding complexity than comparable quantum block codes or previous quantum convolutional codes. Rate-(n-2)/n block stabilizer codes with the same rate and error-correction capability and essentially the same decoding algorithms are derived from these convolutional codes via tail-biting.Comment: 30 pages. Submitted to IEEE Transactions on Information Theory. Minor revisions after first round of review

On the generalization of the construction of quantum codes from Hermitian self-orthogonal codes

Many q-ary stabilizer quantum codes can be constructed from Hermitian self-orthogonal q2-ary linear codes. This result can be generalized to q2m-ary linear codes, m>1. We give a result for easily obtaining quantum codes from that generalization. As a consequence we provide several new binary stabilizer quantum codes which are records according to Grassl (Bounds on the minimum distance of linear codes, http://www.codetables.de, 2020) and new q-ary ones, with q≠2, improving others in the literature.Funding for open access charge: CRUE-Universitat Jaume

Commutative association schemes

Association schemes were originally introduced by Bose and his co-workers in the design of statistical experiments. Since that point of inception, the concept has proved useful in the study of group actions, in algebraic graph theory, in algebraic coding theory, and in areas as far afield as knot theory and numerical integration. This branch of the theory, viewed in this collection of surveys as the "commutative case," has seen significant activity in the last few decades. The goal of the present survey is to discuss the most important new developments in several directions, including Gelfand pairs, cometric association schemes, Delsarte Theory, spin models and the semidefinite programming technique. The narrative follows a thread through this list of topics, this being the contrast between combinatorial symmetry and group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes (based on group actions) and its connection to the Terwilliger algebra (based on combinatorial symmetry). We propose this new role of the Terwilliger algebra in Delsarte Theory as a central topic for future work.Comment: 36 page

Quantum Error-Control Codes

The article surveys quantum error control, focusing on quantum stabilizer codes, stressing on the how to use classical codes to design good quantum codes. It is to appear as a book chapter in "A Concise Encyclopedia of Coding Theory," edited by C. Huffman, P. Sole and J-L Kim, to be published by CRC Press