Locally recoverable code: Difference between revisions

Revision as of 20:57, 8 December 2024

Revision as of 20:59, 8 December 2024

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==Definition==

Let <math>C</math> be a <math>[n, k, d]_{q}</math> [[linear code]]. For <math>i \in \{1, \ldots, n\}</math>, let us denote by <math>r_{i}</math> the minimum number of other [[Coordinate system|coordinates]] we have to look at to recover an erasure in [[Coordinate system|coordinate]] <math>i</math>. The number <math>r_i</math> is said to be the ''locality of the <math>i</math>-th [[Coordinate system|coordinate]]'' of the code. The ''locality'' of the code is defined as <mathdiv displaystyle=block"text-align: center;"><math>r = \max\{r_i \mid i \in \{1, \ldots, n\}\}.</math></div>

An <math>[n, k, d, r]_{q}</math> ''locally recoverable code'' (LRC) is an <math>[n, k, d]_q</math> [[linear code]] <math>C \in \mathbb F_q^n</math> with locality <math>r</math>.