This document is meant to be a reference for people who want to understand the work done on implementing Skew Polynomials and Rank Metric Codes in Sage as a part of this GSoC 2016 project. My mentors for this project are Johan Rosenkilde and David Lucas. This post provides a brief mathematical background, summary and status of the tickets related to the project and further links to detailed discussions and the code. It is assumed that the reader is familiar with the basic concepts of abstract algebra, coding theory and Sage itself.
A Skew Polynomial is given by the equation:
where the coefficients and is a formal variable. Let be a commutative ring equipped with an automorphism . Addition between two skew polynomials is defined by the usual addition operation and the modified multiplication is defined by the rule for all in . Skew polynomials are thus non-commutative and the degree of a product is equal to the sum of the degrees of the factors. Then the ring of skew polynomials can be written as .
Linear Rank Metric Codes are linear error correcting codes over the rank-metric and not the Hamming metric. The distance between two matrices is defined as the rank of the difference between them. And this forms a metric. They are a way to build codes that can detect and correct multiple random rank errors. Gabidulin Codes are an example of linear rank metric codes.
A Gabidulin Code over of length (at most ) and dimension (at most ) is the set of all codewords, that are the evaluation of a skew polynomial belonging to the skew polynomial constructed over the base ring and the twisting automorphism .
where the fixed evaluation points are linearly independent over . These codes are the rank-metric equivalent of Reed Solomon Codes.
During the community bonding period, we started with a small fix to the
wtdist_gap method from the module
linear_code.py. This involved improving its documentation, changing to a better name and making the method private. We also opened a ticket for a new abstract class for Golay Codes. However, there was not enough time to finish this ticket since the coding period started.
David had already implemented a class to manage representation of elements of a field extension into a smaller field. We went through the review checklist in Sage. This has now been merged into Sage.
Xavier Caruso had authored a ticket on Skew Polynomials about four years ago. This was not reviewed at the time. And the category framework in Sage underwent massive restructuring and change since then. As a result, this ticket was very incompatible. We first started by resolving the several hundred doctest errors some of which were due to deprecated syntax errors while others were more technical. This ticket was the most difficult and complicated module and it took a lot longer than originally anticipated. Debugging the code ended up breaking Sage several times before all originally present doctest errors were resolved. Once the ticket started working, a lot of the methods and classes still did not have proper documentation and/or tests and we added those. Since this was a huge ticket, we then decided to break it up into five smaller tickets based on the content and dependencies to make the work more manageable. We majorly refactored the code in the first ticket to make it cleaner and then added methods such as operator evaluation, interpolation, minimum vanishing polynomial, etc to the appropriate various classes. For the purposes of the other targets for the GSoC project, we practically needed only the first of the five tickets (mentioned above) which dealt with the basic implementation of skew polynomials and skew polynomial rings. So the primary focus has been to get that merged. The second ticket based on skew polynomials over finite fields has also been worked upon. The remaining three tickets were more or less lifted directly from the original skew polynomial ticket and no effort has been made yet to accommodate for changes to these.
We then created a new abstract base class for Gabidulin Codes and implemented Generator Matrix based and Skew Polynomial Evaluation based encoders for it. This module also included a Gao Decoder for the Gabidulin Code and a Channel class to introduce random rank errors. Along with all of this, basic getter methods and methods to compute some basic combinatorial properties were also implemented.
Lastly, we worked on a new base class called AbstractLinearRankMetricCode that inherits from the AbstractLinearCode class and from which Gabidulin Code inherits. This class provides methods that are common to any Linear Rank Metric Code so as to facilitate the creation of other such codes. It also deletes those methods from the general linear code class that are not valid in the case of rank metric codes.
Current Status: The tickets related to skew polynomials, Gabidulin Codes and AbstractLinearRankMetricCode are currently still open. We believe that the skew polynomial ticket is almost ready to be merged into Sage. It has been well tested and we believe it is in stable condition. The Gabidulin Code ticket is not as rigorously tested and the Gao decoder in particular needs to be examined further. The completion of this ticket depends on the AbstractLinearRankMetricCode class which requires also requires some work.
I am currently attending Sage Days 75 in Paris to which I was invited by my mentors. We have coding sprints all week long and we aim to wrap up all of the starred tickets (see below) this week. The next section provides links to more elaborate details about the work done during the project.
All relevant tickets, bug reports and code commits can be found below:
- #20565 – Fix LinearCode.wtdist_gap method (enhancement)
- #20787* – A class to manage Golay Codes (new)
- #20284 – A class to manage representations of elements of field extensions (review)
- #13215* – Skew polynomials (enhancement)
- #21088* – A class for skew polynomials over finite fields (enhancement)
- #21131* – Interpolation/evaluation methods for skew polynomials (content)
- #21259* – Karatsuba-based methods for skew polynomials (content)
- #21262* – Center-related content for skew polynomials (content)
- #21264* – Factoring/irreducibility methods for skew polynomials (content)
- #20970* – A class for Gabidulin codes (feature)
- #21226* – An abstract class for rank-metric codes (feature)
Tickets in green represent merged code, tickets in blue represent mature code and the ones in brown represent immature tickets.
The daily project progress log that provides further details about the project can be found at the following link:
The blog posts related to Sage can be found here: