Today we will continue to discuss decoding algorithms for linear error-correcting codes. In order to use the algorithms we discuss, we’ll need to understand the concept of a coset of a subspace. We’ll then use cosets to understand standard array decoding and syndrome decoding for linear codes.
Before our class meeting, you should use the resources below to be able to learn the following. You should be reasonably fluent with these; we’ll answer some questions on them in class but not reteach them in detail.
Let \(V\) be a vector space over a field \(F\text{,}\) and let \(W\) be a subspace of \(V\text{.}\) For any \(v\in V\text{,}\) the coset of \(v\) modulo \(W\) is
