Coding Theory and Algebraic Combinatorics

This chapter introduces and elaborates on the fruitful interplay of coding theory and algebraic combinatorics, with most of the focus on the interaction of codes with combinatorial designs, finite geometries, simple groups, sphere packings, kissing numbers, lattices, and association schemes. In particular, special interest is devoted to the relationship between codes and combinatorial designs. We describe and recapitulate important results in the development of the state of the art. In addition, we give illustrative examples and constructions, and highlight recent advances. Finally, we provide a collection of significant open problems and challenges concerning future research.Comment: 33 pages; handbook chapter, to appear in: "Selected Topics in Information and Coding Theory", ed. by I. Woungang et al., World Scientific, Singapore, 201

Towards a collocation writing assistant for learners of Spanish

This paper describes the process followed in creating a tool aimed at helping learners produce collocations in Spanish. First we present the Diccionario de colocaciones del español (DiCE), an online collocation dictionary, which represents the first stage of this process. The following section focuses on the potential user of a collocation learning tool: we examine the usability problems DiCE presents in this respect, and explore the actual learner needs through a learner corpus study of collocation errors. Next, we review how collocation production problems of English language learners can be solved using a variety of electronic tools devised for that language. Finally, taking all the above into account, we present a new tool aimed at assisting learners of Spanish in writing texts, with particular attention being paid to the use of collocations in this language

kk-Schur functions and affine Schubert calculus

This book is an exposition of the current state of research of affine Schubert calculus and $k$-Schur functions. This text is based on a series of lectures given at a workshop titled "Affine Schubert Calculus" that took place in July 2010 at the Fields Institute in Toronto, Ontario. The story of this research is told in three parts: 1. Primer on $k$-Schur Functions 2. Stanley symmetric functions and Peterson algebras 3. Affine Schubert calculusComment: 213 pages; conference website: http://www.fields.utoronto.ca/programs/scientific/10-11/schubert/, updates and corrections since v1. This material is based upon work supported by the National Science Foundation under Grant No. DMS-065264

Connectivity for bridge-alterable graph classes

A collection of graphs is called bridge-alterable if, for each graph G with a bridge e, G is in the class if and only if G-e is. For example the class of forests is bridge-alterable. For a random forest $F_n$ sampled uniformly from the set of forests on vertex set {1,..,n}, a classical result of Renyi (1959) shows that the probability that $F_n$ is connected is $e^{-1/2 +o(1)}$. Recently Addario-Berry, McDiarmid and Reed (2012) and Kang and Panagiotou (2013) independently proved that, given a bridge-alterable class, for a random graph $R_n$ sampled uniformly from the graphs in the class on {1,..,n}, the probability that $R_n$ is connected is at least $e^{-1/2 +o(1)}$. Here we give a more straightforward proof, and obtain a stronger non-asymptotic form of this result, which compares the probability to that for a random forest. We see that the probability that $R_n$ is connected is at least the minimum over $\frac25 n < t \leq n$ of the probability that $F_t$ is connected.Comment: Amplified the discussion on raising the lower bound 2/5 to 1/