PROGRAM

SC 1 - GROUP ALGEBRAS IN CODING THEORY - C. Polcino Milies and M. Guerreiro
In the first week, we present an introduction to the subject covering some of the important results that can be applied in this contexto, starting with the most basic facts. We begin with the famous theorem of Maschke and use Wedderburn's Theorem to describe the structure of group algebras in the semisimple case and its relation to primitive idempotents. We consider splitting fields and a Theorem of R. Brauer then study the theorem of Berman and Witt that gives the number of simple componentes in the semisimple case.
In the late sixties, S.D Berman [1] and, independently F.J. MacWillims [5], introduced the idea of a group code, de_ned as an ideal of a finite group algebra. In the second week we construct idempotens for abelian codes, always using the structure of subgroups of the underlying group. In some cases, it is possible to compute the parameters of the codes, and bases, using the group algebra structure. The construction of idempotents may also be extended to some non-abelian codes defined from dihedral and quaternion groups We finish mentioning some further developments on codes over rings.
Course Notes:

1. Group Algebras in Coding Theory (file 1)
2. Group Algebras in Coding Theory (file 2)

SC 2 - CHARACTER-THEORETIC TOOLS FOR STUDYING LINEAR CODES OVER RINGS AND MODULES - Jay Wood
Linear codes were originally developed in the context of vector spaces over a finite field. In the past 25 years there has been increasing interest in understanding linear codes in the contexto of modules over a finite ring. Several fundamental results over finite fields due to MacWilliams have character-theoretic proofs that generalize over suitable rings and modules. This mini-course will provide a careful development of the necessary character theory and its applications to linear codes.

1. Linear Codes over Finite Fields
2. Additive Codes and Characters
3. Duality for linear codes
4. Self-duality for linear codes over modules
5. Exercise Session
6. MacWilliams extension theorem and converse
7. MacWilliams extension theorem for other weights
8. Isometries of additive codes
9. Using semigroup rings
10. Extension Problem for general weights

SC 3 - NUMERICAL SEMIGROUPS IN CODING THEORY – Maria Bras- Amorós
We intend to introduce the theory of numerical semigroups, its main results and applications to coding theory. Numerical semigroups appear naturally in coding theory, e.g. in the context of AG codes supported at one point, where Weierstrass semigroups can be used to find bounds for the minimum distance of the code. We will discuss this and other bounds coming from semigroup theory. We also want to present some problems related to the classification, characterization and counting of semigroups.
Course Notes:

• Day 1: Numerical semigroups and codes
• Day 2: Classification, Characterization and Counting of Semigroups
• Day 3: Numerical Semigroups and Alegebraic Geometry Codes
• Day 4: Semigroup Ideals and Generalized Hamming Weights

1. M. Bras-Amorós: Numerical Semigroups and Codes,
   Chapter 5 of Algebraic Geometry Modeling in Information Theory , E. Martinez-Moro (ed.), World Scientific, 2013. ISBN: 978-981-4335-75-1.
2. M. Bras-Amorós, A. Vico-Oton: On the Geil-Matsumoto Bound and the Length of AG codes
   Designs, Codes and Cryptography, Springer, vol. 70, n. 1-2, pp. 117-125, January 2014. ISSN: 0925-1022.
3. M. Bras-Amorós: The Ordinarization Transform of a Numerical Semigroup and Semigroups with a Large Number of Intervals,
   Journal of Pure and Applied Algebra, Elsevier, vol. 216, n. 11, pp. 2507-2518, November 2012. ISSN:0022-4049.
4. M. Bras-Amorós, K. Lee, A. Vico-Oton: New Lower Bounds on the Generalized Hamming Weights of AG Codes
   IEEE Transactions on Information Theory, vol. 60, n. 10, pp. 5930-5937, October 2014. ISSN: 0018-9448.
5. M. Bras-Amorós: Pythagorean moulds, fractal moulds on the golden ratio, half-closed-pipe admissible moulds, and the mathematical emergence of the well tempered harmonic semigroup, submitted, 2017.

SC 4 - APPLICATIONS OF RESULTS FROM COMMUTATIVE ALGEBRA TO THE STUDY OF CERTAIN EVALUATION CODES - Cícero Carvalho
We will start by introducing affine algebraic varieties and Reed-Muller type evaluation codes defined over them. Then we will show how the Hilbert function of the ideal of the variety can be used to determine the dimension of such codes. Following, we show how to use tools from Gröbner basis theory to determine the minimum distance of such codes (or lower bounds for it). We finish the course by showing how to refine this technique and calculate some of the higher Hamming weights of those codes.
Course Notes:
Applications of results from commutative algebra to the study of certain evaluation codes

SC 5 - CONVOLUTIONAL CODES - Diego Napp
The aim of this minicourse is to introduce the class of convolutional codes, their properties and their use in practice. These codes are mathematically more involved than the standard block codes, since the data is seen as a sequence. Although they split the data into blocks of a fixed rate as block codes do, the relative position of each block in the sequence is taken into account. At the end of the course we will present some of the most fascinating open problems in the design of these codes.
Course Notes:

• Day 1: Basics, From block codes to convolutional codes, Encoders
• Day 2: Distance of convolutional codes
• Day 3: Decoding of convolutional codes
• Day 4: Convolutional codes for Network coding
• Day 5: 2D Convolutional Codes

SC 6 - SOME ALGORITHMS FROM DEFINING SETS IN ABELIAN CODES - Juan Jacobo Simón Pinero.
We present some algorithms to obtain two fundamental ingredients for decoding purposes in abelian codes: information sets and the minimum distance. Our algorithms are written only in terms of the defining set of a semisimple abelian code. We then apply this results to the geometrical properties of the information sets to obtain suficiente conditions for na error correcting abelian code.
Course Notes:

1. From ds-bounds for cyclic codes to true distance for abelian codes
2. Information sets from defining sets in abelian codes
3. Information sets in abelian codes: defining sets and Groebner basis
4. Partial Permutation Decoding for abelian codes
5. Cyclic and BCH codes whose minimum distance equals their maximum BCH bound