354 research outputs found
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
Arithmetic Circuit Lower Bounds via MaxRank
We introduce the polynomial coefficient matrix and identify maximum rank of
this matrix under variable substitution as a complexity measure for
multivariate polynomials. We use our techniques to prove super-polynomial lower
bounds against several classes of non-multilinear arithmetic circuits. In
particular, we obtain the following results :
As our main result, we prove that any homogeneous depth-3 circuit for
computing the product of matrices of dimension requires
size. This improves the lower bounds by Nisan and
Wigderson(1995) when .
There is an explicit polynomial on variables and degree at most
for which any depth-3 circuit of product dimension at most
(dimension of the space of affine forms feeding into each
product gate) requires size . This generalizes the lower bounds
against diagonal circuits proved by Saxena(2007). Diagonal circuits are of
product dimension 1.
We prove a lower bound on the size of product-sparse
formulas. By definition, any multilinear formula is a product-sparse formula.
Thus, our result extends the known super-polynomial lower bounds on the size of
multilinear formulas by Raz(2006).
We prove a lower bound on the size of partitioned arithmetic
branching programs. This result extends the known exponential lower bound on
the size of ordered arithmetic branching programs given by Jansen(2008).Comment: 22 page
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