1.
T/F: The repetition code \(C_{3,\text{rep}}\) that consists of repeating a four bit message three times each is a code over the binary alphabet.
Section Daily Prep 2
Today you will learn the basic definitions in coding theory and begin practicing understanding the distance of a code and the decoding process.
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: alphabet, block length, code, codeword, dimension, rate. Hamming distance, and minimum distance of a code.
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Compute the Hamming distance of a pair of vectors and the minimum distance of a given code.
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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State the definition and give examples of: encoding, decoding, \(t\)-error correcting code, \(t\)-error detecting code.
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Given a code and a received vector, decode the vector using nearest-neighbor decoding.
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State and explain the following mathematical result(s): Proposition 1.4.2 (relation between minimum distance and error-detecting/correcting capabilities of a 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 1.2-1.4, pp. 5-15
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Roth, Sections 1.2-1.5 pp. 3-15
Exercises Exercises
2.
What is the block length of the repetition code \(C_{3,\text{rep}}\) that consists of repeating a four bit message three times each?
3.
What is the dimension of the repetition code \(C_{3,\text{rep}}\) that consists of repeating a four bit message three times each?
4.
What is the rate of the repetition code \(C_{3,\text{rep}}\) that consists of repeating a four bit message three times each?
5.
In your own words, what is the Hamming distance between two vectors of the same length over a given alphabet?
6.
What is the Hamming distance between the words \((1,0,0,0,1,1,1)\) and \((1,1,0,0,0,1,1)\text{?}\)
7.
What is the minimum distance of the repetition code \(C_{3,\text{rep}}\) that consists of repeating a four bit message three times each?
8.
What questions do you have about the material for today?
