Feng-Rao decoding of primary codes

We show that the Feng-Rao bound for dual codes and a similar bound by Andersen and Geil [H.E. Andersen and O. Geil, Evaluation codes from order domain theory, Finite Fields Appl., 14 (2008), pp. 92-123] for primary codes are consequences of each other. This implies that the Feng-Rao decoding algorithm can be applied to decode primary codes up to half their designed minimum distance. The technique applies to any linear code for which information on well-behaving pairs is available. Consequently we are able to decode efficiently a large class of codes for which no non-trivial decoding algorithm was previously known. Among those are important families of multivariate polynomial codes. Matsumoto and Miura in [R. Matsumoto and S. Miura, On the Feng-Rao bound for the L-construction of algebraic geometry codes, IEICE Trans. Fundamentals, E83-A (2000), pp. 926-930] (See also [P. Beelen and T. H{\o}holdt, The decoding of algebraic geometry codes, in Advances in algebraic geometry codes, pp. 49-98]) derived from the Feng-Rao bound a bound for primary one-point algebraic geometric codes and showed how to decode up to what is guaranteed by their bound. The exposition by Matsumoto and Miura requires the use of differentials which was not needed in [Andersen and Geil 2008]. Nevertheless we demonstrate a very strong connection between Matsumoto and Miura's bound and Andersen and Geil's bound when applied to primary one-point algebraic geometric codes.Comment: elsarticle.cls, 23 pages, no figure. Version 3 added citations to the works by I.M. Duursma and R. Pellikaa

Qudit Colour Codes and Gauge Colour Codes in All Spatial Dimensions

Two-level quantum systems, qubits, are not the only basis for quantum computation. Advantages exist in using qudits, d-level quantum systems, as the basic carrier of quantum information. We show that color codes, a class of topological quantum codes with remarkable transversality properties, can be generalized to the qudit paradigm. In recent developments it was found that in three spatial dimensions a qubit color code can support a transversal non-Clifford gate, and that in higher spatial dimensions additional non-Clifford gates can be found, saturating Bravyi and K\"onig's bound [Phys. Rev. Lett. 110, 170503 (2013)]. Furthermore, by using gauge fixing techniques, an effective set of Clifford gates can be achieved, removing the need for state distillation. We show that the qudit color code can support the qudit analogues of these gates, and show that in higher spatial dimensions a color code can support a phase gate from higher levels of the Clifford hierarchy which can be proven to saturate Bravyi and K\"onig's bound in all but a finite number of special cases. The methodology used is a generalisation of Bravyi and Haah's method of triorthogonal matrices [Phys. Rev. A 86 052329 (2012)], which may be of independent interest. For completeness, we show explicitly that the qudit color codes generalize to gauge color codes, and share the many of the favorable properties of their qubit counterparts.Comment: Authors' final cop

High-entropy dual functions over finite fields and locally decodable codes

We show that for infinitely many primes p, there exist dual functions of order k over Fnp that cannot be approximated in L∞-distance by polynomial phase functions of degree k−1. This answers in the negative a natural finite-field analog of a problem of Frantzikinakis on L∞-approximations of dual functions over N (a.k.a. multiple correlation sequences) by nilsequences

FRUSTRATION: PHYSICO-CHEMICAL PREREQUISITES FOR THE CONSTRUCTION OF A SYNTHETIC CELL

To construct a synthetic cell we need to understand the rules that permit life. A central idea in modern biology is that in addition to the four entities making reality, matter, energy, space and time, a fifth one, information, plays a central role. As a consequence of this central importance of the management of information, the bacterial cell is organised as a Turing machine, where the machine, with its compartments defining an inside and an outside and its metabolism, reads and expresses the genetic program carried by the genome. This highly abstract organisation is implemented using concrete objects and dynamics, and this is at the cost of repeated incompatibilities (frustration), which need to be sorted out by appropriate «patches». After describing the organisation of the genome into the paleome (sustaining and propagating life) and the cenome (permitting life in context), we describe some chemical hurdles that the cell as to cope with, ending with the specific case of the methionine salvage pathwa

Court of Appeal found no love for Topshop tank: the image right that dare not speak its name

This article contains an analysis of the first instance and appeal decisions of the "Rihanna" case. In particular, the authors consider the substantive law of passing off in the context of the unauthorised use of a celebrity’s image on a Topshop tank vest top. This is followed by a discussion of the consequences of the case for celebrities, consumers and stakeholders in the entertainment and fashion industries

Ethical challenges in argumentation and dialogue in a healthcare context.

As the average age of the population increases, so too do the number of people living with chronic illnesses. With limited resources available, the development of dialogue-based e-health systems that provide justified general health advice offers a cost-effective solution to the management of chronic conditions. It is however imperative that such systems are responsible in their approach. We present in this paper two main challenges for the deployment of e-health systems, that have a particular relevance to dialogue and argumentation: collecting and handling health data, and trust. For both challenges, we look at specific issues therein, outlining their importance in general, and describing their relevance to dialogue and argumentation. Finally, we go on to propose six recommendations for handling these issues, towards addressing the main challenges themselves, that act both as general advice for dialogue and argumentation research in the e-health domain, and as a foundation for future work on this topic