# Quantum Error Correction Via Codes Over Gf(4)

This produces a linear code, with **|Aut(G4)| =** 17280. (In generalthe code is linear precisely when all the elementary divisors are equal to x2+ x + 1.)(b) (x2+ x + 1)2, The image of the map is similarly a binary cyclic codehp(x)i. AAECC aims to encourage cross-fertilization between algebraic methods and their applications in computing and communications. For if D is an (n, 2n+k) linear code, and D′= hD, vi is an(n, 2n+k+1) additive code with minimal distance d, then it is easy to show that the linear have a peek at this web-site

This is the ﬁrst occasion when an impure13 co de exists but a p ure one does not.A second [[6, 1, 3]] code, also impure not equivalent to the ﬁrst, is Rains and P. They also provide the means of defining orthogonality between vectors (zero inner product). The work of the secondauthor was performed while he was with the Institute for Defense Analyses, Princeton, NJ, USA. 1. https://arxiv.org/abs/quant-ph/9608006

Gulliver of Carleton University (Canada) and theUniversity of Canterbury (New Zealand) has extensively studied quasicyclic codes over smallﬁelds [39]. Let ρ be the composition of the natural mapfrom C⊥2to C⊥2/C2with any inner-product preserving map from C⊥2/C2to GF (4)k2. The system returned: (22) Invalid argument The remote host or network may be down. In the classical theory the generalizations of Hammingco des that correct multiple errors are known as BCH cod es [54].

The last ﬁve examples in Table II were found by him. Djordjevic IEEE implementation input introduced Lett matrix representation measurement minimum distance nonbinary obtained optical orthogonal output parity-check matrix phase photon Phys polynomial probability QECC quantum channel quantum circuit quantum codes quantum 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 qubitsDiscover the world's research11+ million Anyelement outside S⊥negates the value of the character at each element of S with which itanticommutes.

Self-dual codesIn this section we study [[n, 0, d]] quantum-error-correcting codes and their associated(n, 2n) self-dual codes C. A pure [[n, n − 2d + 2, d]] code has parameters [[n, n, 1]] (n ≥ 1), [[n, n − 2, 2]](n even ≥ 2), [[5, 1, 3]] or [[6, We assume n is odd.Theorem 13. http://dl.acm.org/citation.cfm?id=2265472 See all ›474 CitationsSee all ›91 ReferencesShare Facebook Twitter Google+ LinkedIn Reddit Download Full-text PDFQuantum error correction via codes over GF(4)Article (PDF Available) in IEEE Transactions on Information Theory 44(4):1369 - 1387 · August 1998 with 49

Terms of Usage Privacy Policy Code of Ethics Contact Us Useful downloads: Adobe Reader QuickTime Windows Media Player Real Player Did you know the ACM DL App is Originally the acronym AAECC meant “Applied Algebra and Error-Correcting Codes”. The applications orientation is towards both theoretical and practical error-correction coding, and, since AAECC 13 (Hawaii, 1999), towards cryptography. Deﬁne a new code Cmto consist ofall vectors v1v2, where v1∈ Cm−2and φ(v1) = v2+ G′m.

Of course in view of Theorem6(c)this also gives bounds on the minimal distance of any pure [[n, k, d]] code.We see from that table that the bound in Theorem17 for even http://www.researchgate.net/publication/3079541_Quantum_error_correction_via_codes_over_GF(4) It is easily veriﬁed that C is additive and that C ⊆ C⊥= ¯ωC⊥1+ ωC2.Another construction based on binary codes due to Gottesman [36] can be generalized asfollows.Theorem 10. Then we obtain a [[25, 1, 9]] code for which the associated(25, 224) an d (25, 226) linear codes have the generator matrices shown in Fig. 1. While there are many similarities, there are also substantialdiﬀerences between the two.

I n particular A56= 0, and so d = 5.(ii) No [[18, 12, 3]] code exists. Check This Out Fu rthermore, this subgroup C has the follow ing characterizationin terms of the error group : an error e is in C precisely when translation by e takes codewordsto codewords and Jungnickel, H. SloaneAT&T Labs - Research, Florham Park, New Jersey 07932-0971Mar. 5 1998ABSTRACTThe problem of ﬁnding quantum-error-cor recting codes is transformed into th e problem ofﬁnding additive codes over the ﬁeld GF (4)

Then there is a quantum-error-correcting code mapping k qubi ts to nqubits which can correct [(d − 1)/2] errors.We will describe such a quantum-error-correcting code by saying it has parameters [[n, If an [[n, k, d]] quantum-error-correcting code exists such that the associated(n, 2n−k) code C contains no vectors of weight 1, then there is a solution to the following set26 of A simple counting argument veriﬁesthat Cmhas the claimed weight distrib ution. Source The subgroup of Gnﬁxing aco de C is its automorphism group Aut(C).

M Rains (2), P. Given two codes [[n1, k1, d1]] and [[n2, k2, d2]] with k2≤ n1we can construct an[[n1+ n2− k2, k1, d]] code, where d ≥ m in{d1, d1+ d2− k2}.14 Proof. It is known [51] that there isno (24, 412) d = 10 classical code over GF (4), but the possibility of a (24, 224) d = 10 additiveself-dual code remains open.

## There exists a (24, 224) d = 8classical code over GF (2), the Golay code, and at least two (24, 312) d = 9 classical codes overGF (3), all meeting the

Then the code obtainedfrom C by deleting the coordinates in S is also self-orthogonal.Proof. Prior to this appointment in August 2006, he was with University of Arizona, Tucson, USA (as a Research Assistant Professor); University of the West of England, Bristol, UK; University of Bristol, If an [[n, k, d]] quantum-error-correcting code exists then there are homogeneouspolynomials W (x, y), W⊥(x, y) and S(x, y) of degree n such that the following conditions hold:W (1, 0) Equivalently, Gnis the wreath product of S3by Sngenerated by permutations of the coordinates and arbitrarypermutations of the nonzero elements of GF (4) in each coordinate.

To appear in IEEE Transactions on Information Theory. Assuming no knowledge of quantum mechanics and written at an intuitive level suitable for the engineer, the book gives all the essential principles needed to design and implement quantum electronic and M. have a peek here A.

Desq, D. This inducesthe symplectic inner product given in (1):((a|b), (a′|b′)) = a · b′+ a′· b ,where we write (a|b) for the image of X(a)Z(b) in¯E. If a pure [[n, k, d]] code exists then k ≤ n − 2d + 2.Proof. Generated Tue, 06 Dec 2016 04:23:32 GMT by s_ac16 (squid/3.5.20)

In all but one of those cases the code can be taken tobe a classical self-dual linear code over GF (4). If κ = 1the code is cyclic. The balance between theoretical and practical is intended to shift regularly; at AAECC-14 the focus was on the theoretical side. He presently directs the Optical Communications Systems Laboratory (OCSL) within the ECE Department at the University of Arizona.

The linear programmin g bound (Theorems21 and22) presented in Section 7 appears to give quite sharp bounds for those codes. Prior to this appointment, he was with University of Arizona, Tucson, USA (as an Assistant Professor and a Research Assistant Professor); University of the West of England, Bristol, UK; University of Analogous to that of Theorem 4 of [24].With the h elp of Theorems 19 and 20 we ﬁnd that the numbers tn(respectively in) ofinequivalent (respectively inequivalent indecomposable) self-dual codes of length Please try the request again.

Similarly, LRis generated by ER, (3) and all matrices diag((−1)φ(v))v∈V,where φ is now any2-valued quadr atic form on V .We also record some further properties of L and LR:• L/hE, ηIi