Incidence structures from the blown-up plane and LDPC codes

In this article, new regular incidence structures are presented. They arise from sets of conics in the affine plane blown-up at its rational points. The LDPC codes given by these incidence matrices are studied. These sparse incidence matrices turn out to be redundant, which means that their number of rows exceeds their rank. Such a feature is absent from random LDPC codes and is in general interesting for the efficiency of iterative decoding. The performance of some codes under iterative decoding is tested. Some of them turn out to perform better than regular Gallager codes having similar rate and row weight.Comment: 31 pages, 10 figure

An upper bound on the number of rational points of arbitrary projective varieties over finite fields

We give an upper bound on the number of rational points of an arbitrary Zariski closed subset of a projective space over a finite field. This bound depends only on the dimensions and degrees of the irreducible components and holds for very general varieties, even reducible and non equidimensional. As a consequence, we prove a conjecture of Ghorpade and Lachaud on the maximal number of rational points of an equidimensional projective variety

Design for (every)one: co-creation as a bridge between universal design and rehabilitation engineering

In this paper the authors describe a general framework for co-designing assistive devices in a horizontal user innovation network [1] by and for disabled users. This framework attempts to identify, share and use “hidden solutions” in rehabilitation contexts and translate them into disruptive assistive devices build with local resources. Within healthcare contexts local solutions are frequently more effective, as they reflect the physical, emotional and cognitive needs of specific patients and engage all the stakeholders in a specific local context. By using an open horizontal innovation network, where assistive devices can be easily shared and physically hacked by other paramedics, general patterns can be detected and translated into standard universal design objects. This generative design thinking approach [2] is more than feasible with digital trends like crowd sourcing, user-generated content and peer production [3]. Cheap and powerful prototyping tools have become easier to use by non-engineers; it turns them into users as well as self manufactures [4]. We discuss the different aspects of this open innovation process within a ‘design for disability’ context and suggest the first steps of an iterative co-design methodology bringing together professional designers, occupational therapists and patients. In this paper the authors sketch the holistic framework which starts with the innovation development and the co-creation process between these disciplines

A Construction of Quantum LDPC Codes from Cayley Graphs

We study a construction of Quantum LDPC codes proposed by MacKay, Mitchison and Shokrollahi. It is based on the Cayley graph of Fn together with a set of generators regarded as the columns of the parity-check matrix of a classical code. We give a general lower bound on the minimum distance of the Quantum code in $\mathcal{O}(dn^2)$ where d is the minimum distance of the classical code. When the classical code is the $[n, 1, n]$ repetition code, we are able to compute the exact parameters of the associated Quantum code which are $[[2^n, 2^{\frac{n+1}{2}}, 2^{\frac{n-1}{2}}]]$.Comment: The material in this paper was presented in part at ISIT 2011. This article is published in IEEE Transactions on Information Theory. We point out that the second step of the proof of Proposition VI.2 in the published version (Proposition 25 in the present version and Proposition 18 in the ISIT extended abstract) is not strictly correct. This issue is addressed in the present versio

Cryptanalysis of McEliece Cryptosystem Based on Algebraic Geometry Codes and their subcodes

We give polynomial time attacks on the McEliece public key cryptosystem based either on algebraic geometry (AG) codes or on small codimensional subcodes of AG codes. These attacks consist in the blind reconstruction either of an Error Correcting Pair (ECP), or an Error Correcting Array (ECA) from the single data of an arbitrary generator matrix of a code. An ECP provides a decoding algorithm that corrects up to $\frac{d^*-1-g}{2}$ errors, where $d^*$ denotes the designed distance and $g$ denotes the genus of the corresponding curve, while with an ECA the decoding algorithm corrects up to $\frac{d^*-1}{2}$ errors. Roughly speaking, for a public code of length $n$ over $\mathbb F_q$, these attacks run in $O(n^4\log (n))$ operations in $\mathbb F_q$ for the reconstruction of an ECP and $O(n^5)$ operations for the reconstruction of an ECA. A probabilistic shortcut allows to reduce the complexities respectively to $O(n^{3+\varepsilon} \log (n))$ and $O(n^{4+\varepsilon})$. Compared to the previous known attack due to Faure and Minder, our attack is efficient on codes from curves of arbitrary genus. Furthermore, we investigate how far these methods apply to subcodes of AG codes.Comment: A part of the material of this article has been published at the conferences ISIT 2014 with title "A polynomial time attack against AG code based PKC" and 4ICMCTA with title "Crypt. of PKC that use subcodes of AG codes". This long version includes detailed proofs and new results: the proceedings articles only considered the reconstruction of ECP while we discuss here the reconstruction of EC

New Identities Relating Wild Goppa Codes

For a given support $L \in \mathbb{F}_{q^m}^n$ and a polynomial $g\in \mathbb{F}_{q^m}[x]$ with no roots in $\mathbb{F}_{q^m}$, we prove equality between the $q$-ary Goppa codes $\Gamma_q(L,N(g)) = \Gamma_q(L,N(g)/g)$ where $N(g)$ denotes the norm of $g$, that is $g^{q^{m-1}+\cdots +q+1}.$ In particular, for $m=2$, that is, for a quadratic extension, we get $\Gamma_q(L,g^q) = \Gamma_q(L,g^{q+1})$. If $g$ has roots in $\mathbb{F}_{q^m}$, then we do not necessarily have equality and we prove that the difference of the dimensions of the two codes is bounded above by the number of distinct roots of $g$ in $\mathbb{F}_{q^m}$. These identities provide numerous code equivalences and improved designed parameters for some families of classical Goppa codes.Comment: 14 page

The role of flow experience in codesigning open-design assistive devices

This paper describes the theoretical framework of an inclusive participatory design approach which leads to qualitative occupational experiences within the field of community-based rehabilitation. The aim is to support voluntarily controlled activities by applying co-construction theories to disabled users and their dynamic environment. The starting point of this open design process is a threefold interaction involving caregivers, patients and occupational therapists within their local product ecology. Co-creation is used as a set of iterative techniques to steer the patient towards flow experiences. Do-it-Yourself is consecutively applied as physical prototyping, communication language and personal manufacturing process. By implementing this active engagement process disabled people and their carers become conscious actors in providing collaborative maintenance of their own physical, mental and social well-being

Cryptanalysis of public-key cryptosystems that use subcodes of algebraic geometry codes

We give a polynomial time attack on the McEliece public key cryptosystem based on subcodes of algebraic geometry (AG) codes. The proposed attack reposes on the distinguishability of such codes from random codes using the Schur product. Wieschebrink treated the genus zero case a few years ago but his approach cannot be extent straightforwardly to other genera. We address this problem by introducing and using a new notion, which we call the t-closure of a code