Locally testable code

Key takeaways

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

Reference excerpt

A locally testable code is a type of error-correcting code for which it can be determined if a string is a word in that code by looking at a small (frequently constant) number of bits of the string. In some situations, it is useful to know if the data is corrupted without decoding all of it so that appropriate action can be taken in response. For example, in communication, if the receiver encounters a corrupted code, it can request the data be re-sent, which could increase the accuracy of said data. Similarly, in data storage, these codes can allow for damaged data to be recovered and rewritten properly. In contrast, locally decodable codes use a small number of bits of the codeword to probabilistically recover the original information. The fraction of errors determines how likely it is that the decoder correctly recovers the original bit; however, not all locally decodable codes are locally testable. Clearly, any valid codeword should be accepted as a codeword, but strings that are not codewords could be only one bit off, which would require many (certainly more than a constant number) probes. To account for this, testing failure is only defined if the string is off by at least a set fraction of its bits. This implies words of the code must be longer than the input strings by adding some redundancy.

Definition To measure the distance between two strings, the Hamming distance is used

Δ ( x , y ) = | { i : x i ≠ y i } | {\displaystyle \Delta (x,y)=|\{i:x_{i}\neq y_{i}\}|}

The distance of a string w {\displaystyle w} from a code C : { 0 , 1 } k → { 0 , 1 } n {\displaystyle C:\{0,1\}^{k}\to \{0,1\}^{n}} is computed by

Δ ( w , C ) = min x { Δ ( w , C ( x ) ) } {\displaystyle \Delta (w,C)=\min _{x}\{\Delta (w,C(x))\}}

Relative distances are computed as a fraction of the number of bits

δ ( x , y ) = Δ ( x , y ) / n and δ ( w , C ) = Δ ( w , C ) / n {\displaystyle \delta (x,y)=\Delta (x,y)/n{\text{ and }}\delta (w,C)=\Delta (w,C)/n}

A code C : { 0 , 1 } k → { 0 , 1 } n {\displaystyle C:\{0,1\}^{k}\to \{0,1\}^{n}} is called q {\displaystyle q} -local δ {\displaystyle \delta } -testable if there exists a Turing machine M given random access to an input w {\displaystyle w} that makes at most q {\displaystyle q} non-adaptive queries of w {\displaystyle w} and satisfies the following:

For any x ∈ { 0 , 1 } k {\displaystyle x\in \{0,1\}^{k}} and w = C ( x ) {\displaystyle w=C(x)} , P r [ M w ( 1 k ) = 1 ] = 1 {\displaystyle Pr[M^{w}(1^{k})=1]=1} . In other words, M accepts given access to any codeword of C. For w ∈ { 0 , 1 } n {\displaystyle w\in \{0,1\}^{n}} such that δ ( w , C ) > δ {\displaystyle \delta (w,C)>\delta } , P r [ M w ( 1 k ) = 1 ] ≤ 1 / 2 {\displaystyle Pr[M^{w}(1^{k})=1]\leq 1/2} . M must reject strings δ {\displaystyle \delta } -far from C at least half the time. Also the rate of a code is the ratio between its message length and codeword length

r = | x | | C ( x ) | {\displaystyle r={\frac {|x|}{|C(x)|}}}

Limits It remains an open question whether there are any locally testable codes of linear size, but there are several constructions that are considered "nearly linear":

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Locally testable code

Start with the simplest possible case. Write down what Locally testable code claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In 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 Locally testable code 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 Locally testable code 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 Locally testable code

In research: Locally testable code appears in 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 Locally testable code 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: Locally testable code is common in secondary-school and first-year university syllabi. It links to neighbouring topics Error detection and correction, so understanding it makes those chapters shorter.
In everyday life: Look for Locally testable code 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.

How to study Locally testable code in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Locally testable code means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Locally testable code out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Locally testable code in simple terms?

A locally testable code is a type of error-correcting code for which it can be determined if a string is a word in that code by looking at a small (frequently constant) number of bits of the string. In some situations, it is useful to know if the data is corrupted without decoding all of it so that…

Why does Locally testable code matter?

Because it connects several 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 Locally testable code?

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 Locally testable code.