Today we will discuss 1-error correcting codes in detail. We start with the construction of the binary Hamming codes, which are a family of 1-error correcting linear codes with very nice properties. In class, we’ll talk about using Proposition 2.3.7 to construct general 1-error correcting linear codes from their parity-check matrices, and we’ll also discuss the efficient decoding algorithm for 1-error correcting linear codes. Finally, we’ll generalize the binary Hamming codes to \(q\)-ary Hamming 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.
