New constructions of WOM codes using the Wozencraft ensemble

In this paper we give several new constructions of WOM codes. The novelty in our constructions is the use of the so called Wozencraft ensemble of linear codes. Specifically, we obtain the following results. We give an explicit construction of a two-write Write-Once-Memory (WOM for short) code that approaches capacity, over the binary alphabet. More formally, for every \epsilon>0, 0<p<1 and n =(1/\epsilon)^{O(1/p\epsilon)} we give a construction of a two-write WOM code of length n and capacity H(p)+1-p-\epsilon. Since the capacity of a two-write WOM code is max_p (H(p)+1-p), we get a code that is \epsilon-close to capacity. Furthermore, encoding and decoding can be done in time O(n^2.poly(log n)) and time O(n.poly(log n)), respectively, and in logarithmic space. We obtain a new encoding scheme for 3-write WOM codes over the binary alphabet. Our scheme achieves rate 1.809-\epsilon, when the block length is exp(1/\epsilon). 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 2-write WOM codes. We then give an application of existing constructions of extractors to the problem of designing encoding schemes for memory with defects.Comment: 19 page

A Generalized Sylvester-Gallai Type Theorem for Quadratic Polynomials

In this work we prove a version of the Sylvester-Gallai theorem for quadratic polynomials that takes us one step closer to obtaining a deterministic polynomial time algorithm for testing zeroness of ?^{[3]}???^{[2]} circuits. Specifically, we prove that if a finite set of irreducible quadratic polynomials ? satisfy that for every two polynomials Q?,Q? ? ? there is a subset ? ? ?, such that Q?,Q? ? ? and whenever Q? and Q? vanish then ?_{Q_i??} Q_i vanishes, then the linear span of the polynomials in ? has dimension O(1). This extends the earlier result [Amir Shpilka, 2019] that showed a similar conclusion when |?| = 1. An important technical step in our proof is a theorem classifying all the possible cases in which a product of quadratic polynomials can vanish when two other quadratic polynomials vanish. I.e., when the product is in the radical of the ideal generated by the two quadratics. This step extends a result from [Amir Shpilka, 2019] that studied the case when one quadratic polynomial is in the radical of two other quadratics