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Category: Algebra

Welcome to GF(4)

Welcome to GF(4)

Everyone has solved some version of a linear system in either high school or college mathematics. If you've been keeping up with some of my other posts on algebra, you know that I'm about to either take something familiar away, or twist it into a different form. This time is no different; we're going to…

A Partition by any Other Name

A Partition by any Other Name

I promise I'm actually a probability theorist, despite many of my posts being algebraic in nature. Algebra, as we've seen in several other posts , elegantly generalizes many things in basic arithmetic, leading to highly lucrative applications in coding theory and data protection.  Some definitions in mathematics may not have obvious "practical use", but turn out to yield theorems and results so…

Reduce the Problem: Permutations and Modulo Arithmetic

Reduce the Problem: Permutations and Modulo Arithmetic

We've all seen permutations before. If you have ten distinct items, and rearrange them on a shelf, you've just performed a permutation. A permutation is actually a function that is performing the arrangement on a set of labeled objects. For simplicity, we can just number the objects and work with permuting the numbers.  (more…)

Theory of Coding, Episode 2: Maximum-Likelihood Decoding

Theory of Coding, Episode 2: Maximum-Likelihood Decoding

The introduction to coding theory in this post will now allow us to explore some more interesting topics in coding theory, courtesy of Pinter's A Book of Abstract Algebra. We'll introduce the notion of a code, informations, and parity check equations. Most communication channels are noisy to some extent, which means that a transmitted codeword may have…

Group Theory, XOR, and Binary Codes: Introducing Coding Theory

Group Theory, XOR, and Binary Codes: Introducing Coding Theory

Binary codes and bit-wise operations are fundamental in computer science. Whatever device you are running today works because of binary codes, and the bit-wise operations AND, OR, NOT, and XOR. (A fun exercise, prove to yourself that each of these rules meets the definition of an operation. If you need a refresher, check out this…

Concatenation as an Operation

Concatenation as an Operation

Mathematics is like any activity, sport, or skill: it must be honed and practiced. With that in mind, I have been bolstering up my abilities in algebra with a fantastic book A Book of Abstract Algebra, by Charles C. Pinter.. As I go through the chapters, I will be posting and discussing selected relevant exercises that…

Like Clockwork: Modulo Addition and Finite Groups of Integers

Like Clockwork: Modulo Addition and Finite Groups of Integers

Modulo arithmetic always scared me in college. (Honestly, it's still not something I can do as easily as integration or spotting pdfs hidden inside icky integrals.) Then when you add abstract algebra (or modern algebra as some call it) and group theory on top of it, the whole topic can become intimidating. Many books on…