Invertible zero-error dispersers and defective memory with stuck-at errors

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.