Rewriting Codes for Flash Memories

Flash memory is a nonvolatile computer memory comprising blocks of cells, wherein each cell can take on
$q$
different values or levels. While increasing the cell level is easy, reducing the level of a cell can be accomplished only by erasing an entire block. Since block erasures are highly undesirable, coding schemes—known as floating codes (or flash codes) and buffer codes—have been designed in order to maximize the number of times that information stored in a flash memory can be written (and rewritten) prior to incurring a block erasure. An
$(n,k,t)_{q}$
flash code
$\BBC$
is a coding scheme for storing
$k$
information bits in
$n$
cells in such a way that any sequence of up to
$t$
writes can be accommodated without a block erasure. The total number of available level transitions in
$n$
cells is
$n(q{-}1)$
, and the write deficiency of
$\BBC$
, defined as
$\delta (\BBC)=n(q{-}1)-t$
, is a measure of how close the code comes to perfectly utilizing all these transitions. In this paper, we show a construction of flash codes with write deficiency
$O(qk\log k)$
if
$q\geqslant\log_{2}k$
, and at most
$O(k\log^{2}k)$
otherwise. An
$(n,r,\ell,t)_{q}$
buffer code is a coding scheme for storing a buffer of
$r~\ell$
-ary symbols such that for any sequence of
$t$
symbols, it is possible to successfully decode the last
$r$
symbols that were written. We improve upon a previous upper bound on the maximum number of writes
$t$
in the case where there is a single cell to store the buffer. Then, we show how to improve a construction by Jiang

that uses multiple cells, where
$n\geqslant 2r$
.