Section Daily Prep 5
Today we introduce linear codes over finite fields and the two main ways of describing them: generator matrices and parity-check matrices. In class, we’ll finish the counting exercise from last times and work on understanding the connection between the parity-check matrix and minimum distance of a linear code.
Subsection Learning Objectives
Subsubsection Basic Learning 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.
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State the definition and give examples of: linear code, generator matrix, parity check matrix.
Subsubsection Advanced Learning Objectives
During our class meeting, we will work on learning the following. Fluency with these is not expected or required before class.
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Count the number of subspaces of a given dimension in \(\F_q^n\text{,}\) the number of vectors in a given subspace, and the number of bases for a given subspace.
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State the definition and give examples of: equivalent codes, systematic codes, systematic form of a generator matrix.
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Convert between generator and parity-check matrices for a systematic linear code.
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State and apply Proposition 2.3.7 to determine minimum distance of a linear code.
Subsection Resources for Learning
Use these resources to prepare for class and answer the questions below.
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Guruswami, Rudra, & Sudan, Sections 2.2-2.3, pp. 33-39
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Roth, Sections 2.1-2.3, pp 26-32
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Vanstone & van Oorschot, Sections 3.1-3.3, pp 45-64
