New quantum codes from self-dual codes over F_4

We present new constructions of binary quantum codes from quaternary linear Hermitian self-dual codes. Our main ingredients for these constructions are nearly self-orthogonal cyclic or duadic codes over F_4. An infinite family of $0$-dimensional binary quantum codes is provided. We give minimum distance lower bounds for our quantum codes in terms of the minimum distance of their ingredient linear codes. We also present new results on the minimum distance of linear cyclic codes using their fixed subcodes. Finally, we list many new record-breaking quantum codes obtained from our constructions.Comment: 16 page

Simple Rate-1/3 Convolutional and Tail-Biting Quantum Error-Correcting Codes

Simple rate-1/3 single-error-correcting unrestricted and CSS-type quantum convolutional codes are constructed from classical self-orthogonal \F_4-linear and \F_2-linear convolutional codes, respectively. These quantum convolutional codes have higher rate than comparable quantum block codes or previous quantum convolutional codes, and are simple to decode. A block single-error-correcting [9, 3, 3] tail-biting code is derived from the unrestricted convolutional code, and similarly a [15, 5, 3] CSS-type block code from the CSS-type convolutional code.Comment: 5 pages; to appear in Proceedings of 2005 IEEE International Symposium on Information Theor

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

Low-complexity quantum codes designed via codeword-stabilized framework

We consider design of the quantum stabilizer codes via a two-step, low-complexity approach based on the framework of codeword-stabilized (CWS) codes. In this framework, each quantum CWS code can be specified by a graph and a binary code. For codes that can be obtained from a given graph, we give several upper bounds on the distance of a generic (additive or non-additive) CWS code, and the lower Gilbert-Varshamov bound for the existence of additive CWS codes. We also consider additive cyclic CWS codes and show that these codes correspond to a previously unexplored class of single-generator cyclic stabilizer codes. We present several families of simple stabilizer codes with relatively good parameters.Comment: 12 pages, 3 figures, 1 tabl

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

A survey of complex generalized weighing matrices and a construction of quantum error-correcting codes

Some combinatorial designs, such as Hadamard matrices, have been extensively researched and are familiar to readers across the spectrum of Science and Engineering. They arise in diverse fields such as cryptography, communication theory, and quantum computing. Objects like this also lend themselves to compelling mathematics problems, such as the Hadamard conjecture. However, complex generalized weighing matrices, which generalize Hadamard matrices, have not received anything like the same level of scrutiny. Motivated by an application to the construction of quantum error-correcting codes, which we outline in the latter sections of this paper, we survey the existing literature on complex generalized weighing matrices. We discuss and extend upon the known existence conditions and constructions, and compile known existence results for small parameters. Some interesting quantum codes are constructed to demonstrate their value.Comment: 33 pages including appendi

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

Quantum stabilizer codes

We study quantum stabilizer codes and their connection to classical block codes. In addition, di erent constructions of quantum stabilizer codes and methods of modifying them are presented. Two-dimensional cyclic codes are recalled and a new method of obtaining quantum codes from 2-D cyclic codes is given. We also present a method of obtaining quantum stabilizer codes using additive codes over F4

On the equivalence of linear cyclic and constacyclic codes

We introduce new sufficient conditions for permutation and monomial equivalence of linear cyclic codes over various finite fields. We recall that monomial equivalence and isometric equivalence are the same relation for linear codes over finite fields. A necessary and sufficient condition for the monomial equivalence of linear cyclic codes through a shift map on their defining set is also given. Moreover, we provide new algebraic criteria for the monomial equivalence of constacyclic codes over $\mathbb{F}_4$. Finally, we prove that if $\gcd(3n,\phi(3n))=1$, then all permutation equivalent constacyclic codes of length $n$ over $\mathbb{F}_4$ are given by the action of multipliers. The results of this work allow us to prune the search algorithm for new linear codes and discover record-breaking linear and quantum codes.Comment: 18 page