## Abstract

Kuznetsov and Tsybakov [11] considered the problem of storing information in a memory where some cells are 'stuck' at certain values. More precisely, For 0 < r,p < 1 we want to store a string z ∈ {0,1} ^{rn} in an n-bit memory x = (x _{1},...,x _{n}) in which a subset S ⊆[n] of size pn are stuck at certain values u _{1},...,u _{pn} and cannot be modified. The encoding procedure receives S, u _{1}, u _{pn} and z and can modify the cells outside of S. The decoding procedure should be able to recover z given x (without having to know S or u _{1},...,u _{pn}). This problem is related to, and harder than, the Write-Once-Memory (WOM) problem. We give explicit schemes with rate r ≥ 1 - p - o(1) (trivially, r ≤ 1 - p is a lower bound). This is the first explicit scheme with asymptotically optimal rate. We are able to guarantee the same rate even if following the encoding, the memory x is corrupted in o(√n) adversarially chosen positions. This more general setup was first considered by Tsybakov [24] (see also [10,8]). and our scheme improves upon previous results. We utilize a recent connection observed by Shpilka [21] between the WOM problem and linear seeded extractors for bit-fixing sources. We generalize this observation and show that memory schemes for stuck-at memory are equivalent to zero-error seedless dispersers for bit-fixing sources. We furthermore show that using zero-error seedless dispersers for affine sources (together with linear error correcting codes with large dual distance) allows the scheme to also handle adversarial errors. It turns out that explicitness of the disperser is not sufficient for the explicitness of the memory scheme. We also need that the disperser is efficiently invertible, meaning that given an output z and the linear equations specifying a bit-fixing/affine source, one can efficiently produce a string x in the support of the source on which the disperser outputs z. In order to construct our memory schemes, we give new constructions of zero-error seedless dispersers for bit-fixing sources and affine sources. These constructions improve upon previous work by [14,6,2,25,13] in that for sources with min-entropy k, they (i) achieve larger output length m = (1 - o(1))·k whereas previous constructions did not, and (ii) are efficiently invertible, whereas previous constructions do not seem to be easily invertible.

Original language | English |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization |

Subtitle of host publication | Algorithms and Techniques - 15th International Workshop, APPROX 2012, and 16th International Workshop, RANDOM 2012, Proceedings |

Pages | 553-564 |

Number of pages | 12 |

DOIs | |

State | Published - 2012 |

Event | 15th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2012 and the 16th International Workshop on Randomization and Computation, RANDOM 2012 - Cambridge, MA, United States Duration: 15 Aug 2012 → 17 Aug 2012 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 7408 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 15th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2012 and the 16th International Workshop on Randomization and Computation, RANDOM 2012 |
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Country/Territory | United States |

City | Cambridge, MA |

Period | 15/08/12 → 17/08/12 |

## ASJC Scopus subject areas

- Theoretical Computer Science
- General Computer Science