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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 ℂ 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 ℂ, defined as δ(ℂ)=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(q k log k) if q ≥ log2 k, and at most O(klog2k) otherwise. An (n,r,l,t)q buffer code is a coding scheme for storing a buffer of r l-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 ≥ 2r.