computer science

Guruswami–Sudan list decoding algorithm

Guruswami–Sudan list decoding algorithm is a computer science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Guruswami–Sudan list decoding algorithm rather than just read about it. In short: In coding theory, list decoding is an alternative to unique decoding of error-correcting codes in the presence of many errors. If a code has relative distance δ {\displaystyle \delta } , then it is possible in principle to recover an encoded message when up to δ / 2 {\displaystyle \delta /2} fraction of the codeword symbols are corrupted.

Key takeaways

Guruswami–Sudan list decoding algorithm belongs to computer science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Guruswami–Sudan list decoding algorithm to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Guruswami–Sudan list decoding algorithm from memory before moving on to harder problems.

Reference excerpt

In coding theory, list decoding is an alternative to unique decoding of error-correcting codes in the presence of many errors. If a code has relative distance δ {\displaystyle \delta } , then it is possible in principle to recover an encoded message when up to δ / 2 {\displaystyle \delta /2} fraction of the codeword symbols are corrupted. But when error rate is greater than δ / 2 {\displaystyle \delta /2} , this will not in general be possible. List decoding overcomes that issue by allowing the decoder to output a short list of messages that might have been encoded. List decoding can correct more than δ / 2 {\displaystyle \delta /2} fraction of errors. There are many polynomial-time algorithms for list decoding. In this article, we first present an algorithm for Reed–Solomon (RS) codes which corrects up to 1 − 2 R {\displaystyle 1-{\sqrt {2R}}} errors and is due to Madhu Sudan. Subsequently, we describe the improved Guruswami–Sudan list decoding algorithm, which can correct up to 1 − R {\displaystyle 1-{\sqrt {R}}} errors. Here is a plot of the rate R and distance δ {\displaystyle \delta } for different algorithms. https://wiki.cse.buffalo.edu/cse545/sites/wiki.cse.buffalo.edu.cse545/files/81/Graph.jpg

Algorithm 1 (Sudan's list decoding algorithm)

Problem statement Input : A field F {\displaystyle F} ; n distinct pairs of elements ( x i , y i ) i = 1 n {\displaystyle {(x_{i},y_{i})_{i=1}^{n}}} in F × F {\displaystyle F\times F} ; and integers d {\displaystyle d} and t {\displaystyle t} . Output: A list of all functions f : F → F {\displaystyle f:F\to F} satisfying

f ( x ) {\displaystyle f(x)} is a polynomial in x {\displaystyle x} of degree at most d {\displaystyle d}

To understand Sudan's Algorithm better, one may want to first know another algorithm which can be considered as the earlier version or the fundamental version of the algorithms for list decoding RS codes - the Berlekamp–Welch algorithm. Welch and Berlekamp initially came with an algorithm which can solve the problem in polynomial time with best threshold on t {\displaystyle t} to be t ≥ ( n + d + 1 ) / 2 {\displaystyle t\geq (n+d+1)/2} . The mechanism of Sudan's Algorithm is almost the same as the algorithm of Berlekamp–Welch Algorithm, except in the step 1, one wants to compute a bivariate polynomial of bounded ( 1 , k ) {\displaystyle (1,k)} degree. Sudan's list decoding algorithm for Reed–Solomon code which is an improvement on Berlekamp and Welch algorithm, can solve the problem with t = ( 2 n d ) {\displaystyle t=({\sqrt {2nd}})} . This bound is better than the unique decoding bound 1 − ( R 2 ) {\displaystyle 1-\left({\frac {R}{2}}\right)} for R < 0.07 {\displaystyle R<0.07} .

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Guruswami–Sudan list decoding algorithm

Start with the simplest possible case. Write down what Guruswami–Sudan list decoding algorithm claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In computer science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Guruswami–Sudan list decoding algorithm before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Guruswami–Sudan list decoding algorithm ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Guruswami–Sudan list decoding algorithm

In research: Guruswami–Sudan list decoding algorithm appears in computer science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Guruswami–Sudan list decoding algorithm in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Guruswami–Sudan list decoding algorithm is common in secondary-school and first-year university syllabi. It links to neighbouring topics Coding theory, so understanding it makes those chapters shorter.
In everyday life: Look for Guruswami–Sudan list decoding algorithm outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Guruswami–Sudan list decoding algorithm in 20 minutes

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Frequently asked questions

What is Guruswami–Sudan list decoding algorithm in simple terms?

Why does Guruswami–Sudan list decoding algorithm matter?

Because it connects several computer science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Guruswami–Sudan list decoding algorithm?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Guruswami–Sudan list decoding algorithm.