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Section Daily Prep 7
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.
Subsection Learning Objectives
Subsubsection Basic Learning Objectives
Objectives
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.
State the definition and give examples of cosets of a subspace.
Subsubsection Advanced Learning Objectives
Objectives
During our class meeting, we will work on learning the following. Fluency with these is not expected or required before class.
Construct a standard array for a given linear code.
State and apply the standard array decoding algorithm for linear codes.
State and apply the syndrome decoding algorithm for linear codes.
Subsection Resources for Learning
Use these resources to prepare for class and answer the questions below.
Roth, Sections 2.4, pp. 32-36
Vanstone & van Oorschot, Sections 3.6, pp 69-78
Axler , Section 3E, pp. 98-100 (reference for cosets)
Subsection Important Terms
Definition 41 . Coset of a Subspace.
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
\begin{equation*}
v+W=\{v+w\mid w\in W\}
\end{equation*}
and we call \(v\) a representative of the coset.
