Recent results in the decoding of Algebraic geometry codes

Objectives: The aim of this study was to examine the relationships between perceived teacher autonomy support versus control and students’ life skills development in PE, and whether students’ basic need satisfaction and frustration mediated these relationships. Design: Cross-sectional study. Method: English and Irish students (N = 407, Mage = 13.71, SD = 1.23) completed measures assessing perceived autonomy-supportive and controlling teaching, basic need satisfaction and frustration (autonomy, competence, and relatedness), and life skills development in PE (teamwork, goal setting, social skills, problem solving and decision making, emotional skills, leadership, time management, and interpersonal communication). Results: On the bright side of Self-Determination Theory (SDT), correlations revealed that perceived teacher autonomy support was positively associated with students’ basic need satisfaction and life skills development in PE. On the dark side of SDT, perceived controlling teaching was positively related to students’ basic need frustration, but not significantly related to their life skills development. Mediational analyses revealed that autonomy and relatedness satisfaction mediated the relationships between perceived teacher autonomy support and students’ development of all eight life skills. Competence satisfaction mediated the relationships between perceived teacher autonomy support and students’ development of teamwork, goal setting, and leadership skills. Conclusions: Our findings indicate that satisfaction of the needs for autonomy, competence, and relatedness are important mechanisms that in part explain the relationships between perceived teacher autonomy support and life skills development in PE. Therefore, teachers may look to promote students’ perceptions of an autonomy-supportive climate that satisfies their three basic needs and helps to develop their life skills

Decoding of Projective Reed-Muller Codes by Dividing a Projective Space into Affine Spaces

A projective Reed-Muller (PRM) code, obtained by modifying a (classical) Reed-Muller code with respect to a projective space, is a doubly extended Reed-Solomon code when the dimension of the related projective space is equal to 1. The minimum distance and dual code of a PRM code are known, and some decoding examples have been represented for low-dimensional projective space. In this study, we construct a decoding algorithm for all PRM codes by dividing a projective space into a union of affine spaces. In addition, we determine the computational complexity and the number of errors correctable of our algorithm. Finally, we compare the codeword error rate of our algorithm with that of minimum distance decoding.Comment: 17 pages, 4 figure

Generalized Berlekamp-Massey Decoding of Algebraic-Geometric Codes up to Half the Feng-Rao Bound

Abstiuct-We treat a general class of algebraic-geometric codes and show how to decode these up to half the Feng-Rao bound, using an extension and modification of the Sakata algorithm. The Sakata algorithm is a generalization to N dimensions of the classical Berlekamp-Massey algorithm. E Index Terms-Decoding, algebraic-geometric codes

List Decoding Algorithm based on Voting in Groebner Bases for General One-Point AG Codes

We generalize the unique decoding algorithm for one-point AG codes over the Miura-Kamiya Cab curves proposed by Lee, Bras-Amor\'os and O'Sullivan (2012) to general one-point AG codes, without any assumption. We also extend their unique decoding algorithm to list decoding, modify it so that it can be used with the Feng-Rao improved code construction, prove equality between its error correcting capability and half the minimum distance lower bound by Andersen and Geil (2008) that has not been done in the original proposal except for one-point Hermitian codes, remove the unnecessary computational steps so that it can run faster, and analyze its computational complexity in terms of multiplications and divisions in the finite field. As a unique decoding algorithm, the proposed one is empirically and theoretically as fast as the BMS algorithm for one-point Hermitian codes. As a list decoding algorithm, extensive experiments suggest that it can be much faster for many moderate size/usual inputs than the algorithm by Beelen and Brander (2010). It should be noted that as a list decoding algorithm the proposed method seems to have exponential worst-case computational complexity while the previous proposals (Beelen and Brander, 2010; Guruswami and Sudan, 1999) have polynomial ones, and that the proposed method is expected to be slower than the previous proposals for very large/special inputs.Comment: Accepted for publication in J. Symbolic Computation. LaTeX2e article.cls, 42 pages, 4 tables, no figures. Ver. 6 added an illustrative example of the algorithm executio

Sub-quadratic Decoding of One-point Hermitian Codes

We present the first two sub-quadratic complexity decoding algorithms for one-point Hermitian codes. The first is based on a fast realisation of the Guruswami-Sudan algorithm by using state-of-the-art algorithms from computer algebra for polynomial-ring matrix minimisation. The second is a Power decoding algorithm: an extension of classical key equation decoding which gives a probabilistic decoding algorithm up to the Sudan radius. We show how the resulting key equations can be solved by the same methods from computer algebra, yielding similar asymptotic complexities.Comment: New version includes simulation results, improves some complexity results, as well as a number of reviewer corrections. 20 page