This report documents the program and the outcomes of Dagstuhl Seminar 11461 “Coding Theory”. A (channel) code is typically a set of vectors of the same length n over a finite alphabet Σ. By choosing a fixed codebook, binary strings of appropriate length are injectively mapped into the elements of the code. These elements are then transmitted over a communications channel which induces errors o...

These are the notes for the 2011 Summer Tutorial on Coding Theory. I have not gone through and given citations or references for all of the results given here, but the presentation relies heavily on two sources, van Lint’s Introduction to Coding Theory and the book of Huffman and Pless Fundamentals of Error-Correcting Codes. I also used course notes written by Sebastian Pancratz from a Part II ...

Acknowledgements Firstly, I would like to thank my supervisor, Dr David R. Kohel whose many helpful comments and patient guidance made this thesis possible. Secondly, I would like to thank David Gruenwald for volunteering his time to proofread this thesis. Almost all of his recommendations were taken on board. This thesis is the better for it. To a less extent, my thanks go to the IT school of ...

By elementary linear algebra involving row operations (replacing row i by row i plus row j), and column exchanges, we can write G as (Ik|A) where A is some k×n−k matrix. Now the matrix H = ( −A In−k ) satisfies GH = 0. Thus every vector y = xG satisfies yH = 0. Furthermore standard linear algebra implies that the space of vectors that satisfies yH = 0 is at most k and so this must be the space ...

In today’s lecture we’ll go over some preliminaries to the coding theory we will need throughout the course. In particular, we will go over the Walsh-Hadamard code and the Blum-Luby-Rubinfeld linearity test. In the following lecture we will use these tools to show that NP ⊆ PCP [poly, O(1)] (recall that our goal is ultimately to bring the polynomial amount of randomness down to a logarithmic am...

Coding theory has played a central role in the theoretical computer science. Computer scientists have long exploited notions, constructions, theorems and techniques of coding theory. More recently, theoretical computer science has also been contributing to the theory of error-correcting codes in particular in making progress on some fundamental algorithmic connections. Here we survey some of th...