New constructions of WOM codes using the wozencraft ensemble

In this paper, we give several new constructions of write-once-memory (WOM) codes. The novelty in our constructions is the use of the so-calledWozencraft ensemble of linear codes. Specifically, we obtain the following results. We give an explicit construction of a two-write WOM code that approaches capacity, over the binary alphabet. More formally, for every ε > 0,0 < p < 1, and n=(1/∈)O(1/pc), we give a construction of a two-write WOM code of length n and capacity H(p) +1-p-∈. Since the capacity of a two-write WOM code is max_p(H(p) +1-p), we get a code that is ∈-close to capacity. Furthermore, encoding and decoding can be done in time O(n ²· poly (logn)) and time O(n· poly (logn)), respectively, and in logarithmic space. In addition, we exhibit an explicit randomized encoding scheme of a two-write capacity-achieving WOM code of block length polynomial in 1/∈ (again, ∈ is the gap to capacity), with a polynomial time encoding and decoding. We obtain a new encoding scheme for three-writeWOM codes over the binary alphabet. Our scheme achieves rate 1.809-∈, when the block length is exp(1/∈. This gives a better rate than what could be achieved using previous techniques. We highlight a connection to linear seeded extractors for bit-fixing sources. In particular, we show that obtaining such an extractor with seed length O(log n) can lead to improved parameters for two-write WOM codes.We then give an application of existing constructions of extractors to the problem of designing encoding schemes for memory with defects.