On the weights of binary irreducible cyclic codes

International audienceThis paper is devoted to the study of the weights of binary irreducible cyclic codes. We start from McEliece's interpretation of these weights by means of Gauss sums. Firstly, a dyadic analysis, using the Stickelberger congruences and the Gross-Koblitz formula, enables us to improve McEliece's divisibility theorem by giving results on the multiplicity of the weights. Secondly, in connection with a Schmidt and White's conjecture, we focus on binary irreducible cyclic codes of index two. We show, assuming the generalized Riemann hypothesis, that there are an infinite of such codes. Furthermore, we consider a subclass of this family of codes satisfying the quadratic residue conditions. The parameters of these codes are related to the class number of some imaginary quadratic number fields. We prove the non existence of such codes which provide us a very elementary proof, without assuming G.R.H, that any two-weight binary irreducible cyclic code c(m,v) of index two with v prime greater that three is semiprimitive

Convolutional encoding of self-dual codes

There exist almost complete convolutional encodings of self-dual codes, i.e., block codes of rate 1/2 with weights w, w = 0 mod 4. The codes are of length 8m with the convolutional portion of length 8m-2 and the nonsystematic information of length 4m-1. The last two bits are parity checks on the two (4m-1) length parity sequences. The final information bit complements one of the extended parity sequences of length 4m. Solomon and van Tilborg have developed algorithms to generate these for the Quadratic Residue (QR) Codes of lengths 48 and beyond. For these codes and reasonable constraint lengths, there are sequential decodings for both hard and soft decisions. There are also possible Viterbi-type decodings that may be simple, as in a convolutional encoding/decoding of the extended Golay Code. In addition, the previously found constraint length K = 9 for the QR (48, 24;12) Code is lowered here to K = 8

A new class of three-weight linear codes from weakly regular plateaued functions

Linear codes with few weights have many applications in secret sharing schemes, authentication codes, communication and strongly regular graphs. In this paper, we consider linear codes with three weights in arbitrary characteristic. To do this, we generalize the recent contribution of Mesnager given in [Cryptography and Communications 9(1), 71-84, 2017]. We first present a new class of binary linear codes with three weights from plateaued Boolean functions and their weight distributions. We next introduce the notion of (weakly) regular plateaued functions in odd characteristic $p$ and give concrete examples of these functions. Moreover, we construct a new class of three-weight linear $p$-ary codes from weakly regular plateaued functions and determine their weight distributions. We finally analyse the constructed linear codes for secret sharing schemes.Comment: The Extended Abstract of this work was submitted to WCC-2017 (the Tenth International Workshop on Coding and Cryptography

Investigation of sequence features of hinge-bending regions in proteins with domain movements using kernel logistic regression

Background: Hinge-bending movements in proteins comprising two or more domains form a large class of functional movements. Hinge-bending regions demarcate protein domains and collectively control the domain movement. Consequently, the ability to recognise sequence features of hinge-bending regions and to be able to predict them from sequence alone would benefit various areas of protein research. For example, an understanding of how the sequence features of these regions relate to dynamic properties in multi-domain proteins would aid in the rational design of linkers in therapeutic fusion proteins. Results: The DynDom database of protein domain movements comprises sequences annotated to indicate whether the amino acid residue is located within a hinge-bending region or within an intradomain region. Using statistical methods and Kernel Logistic Regression (KLR) models, this data was used to determine sequence features that favour or disfavour hinge-bending regions. This is a difficult classification problem as the number of negative cases (intradomain residues) is much larger than the number of positive cases (hinge residues). The statistical methods and the KLR models both show that cysteine has the lowest propensity for hinge-bending regions and proline has the highest, even though it is the most rigid amino acid. As hinge-bending regions have been previously shown to occur frequently at the terminal regions of the secondary structures, the propensity for proline at these regions is likely due to its tendency to break secondary structures. The KLR models also indicate that isoleucine may act as a domain-capping residue. We have found that a quadratic KLR model outperforms a linear KLR model and that improvement in performance occurs up to very long window lengths (eighty residues) indicating long-range correlations. Conclusion: In contrast to the only other approach that focused solely on interdomain hinge-bending regions, the method provides a modest and statistically significant improvement over a random classifier. An explanation of the KLR results is that in the prediction of hinge-bending regions a long-range correlation is at play between a small number amino acids that either favour or disfavour hinge-bending regions. The resulting sequence-based prediction tool, HingeSeek, is available to run through a webserver at hingeseek.cmp.uea.ac.uk

A Method to determine Partial Weight Enumerator for Linear Block Codes

In this paper we present a fast and efficient method to find partial weight enumerator (PWE) for binary linear block codes by using the error impulse technique and Monte Carlo method. This PWE can be used to compute an upper bound of the error probability for the soft decision maximum likelihood decoder (MLD). As application of this method we give partial weight enumerators and analytical performances of the BCH(130,66), BCH(103,47) and BCH(111,55) shortened codes; the first code is obtained by shortening the binary primitive BCH (255,191,17) code and the two other codes are obtained by shortening the binary primitive BCH(127,71,19) code. The weight distributions of these three codes are unknown at our knowledge.Comment: Computer Engineering and Intelligent Systems Vol 3, No.11, 201