Source Polarization

The notion of source polarization is introduced and investigated. This complements the earlier work on channel polarization. An application to Slepian-Wolf coding is also considered. The paper is restricted to the case of binary alphabets. Extension of results to non-binary alphabets is discussed briefly.Comment: To be presented at the IEEE 2010 International Symposium on Information Theory

Channel combining and splitting for cutoff rate improvement

The cutoff rate $R_0(W)$ of a discrete memoryless channel (DMC) $W$ is often used as a figure of merit, alongside the channel capacity $C(W)$. Given a channel $W$ consisting of two possibly correlated subchannels $W_1$, $W_2$, the capacity function always satisfies $C(W_1)+C(W_2) \le C(W)$, while there are examples for which $R_0(W_1)+R_0(W_2) > R_0(W)$. This fact that cutoff rate can be ``created'' by channel splitting was noticed by Massey in his study of an optical modulation system modeled as a $M$'ary erasure channel. This paper demonstrates that similar gains in cutoff rate can be achieved for general DMC's by methods of channel combining and splitting. Relation of the proposed method to Pinsker's early work on cutoff rate improvement and to Imai-Hirakawa multi-level coding are also discussed.Comment: 5 pages, 7 figures, 2005 IEEE International Symposium on Information Theory, Adelaide, Sept. 4-9, 200

Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels

A method is proposed, called channel polarization, to construct code sequences that achieve the symmetric capacity $I(W)$ of any given binary-input discrete memoryless channel (B-DMC) $W$. The symmetric capacity is the highest rate achievable subject to using the input letters of the channel with equal probability. Channel polarization refers to the fact that it is possible to synthesize, out of $N$ independent copies of a given B-DMC $W$, a second set of $N$ binary-input channels $\{W_N^{(i)}:1\le i\le N\}$ such that, as $N$ becomes large, the fraction of indices $i$ for which $I(W_N^{(i)})$ is near 1 approaches $I(W)$ and the fraction for which $I(W_N^{(i)})$ is near 0 approaches $1-I(W)$. The polarized channels $\{W_N^{(i)}\}$ are well-conditioned for channel coding: one need only send data at rate 1 through those with capacity near 1 and at rate 0 through the remaining. Codes constructed on the basis of this idea are called polar codes. The paper proves that, given any B-DMC $W$ with $I(W)>0$ and any target rate $R < I(W)$, there exists a sequence of polar codes $\{{\mathscr C}_n;n\ge 1\}$ such that ${\mathscr C}_n$ has block-length $N=2^n$, rate $\ge R$, and probability of block error under successive cancellation decoding bounded as P_{e}(N,R) \le \bigoh(N^{-\frac14}) independently of the code rate. This performance is achievable by encoders and decoders with complexity $O(N\log N)$ for each.Comment: The version which appears in the IEEE Transactions on Information Theory, July 200

Planar Contact Structures with Binding Number Three

In this article, we find the complete list of all contact structures (up to isotopy) on closed three-manifolds which are supported by an open book decomposition having planar pages with three (but not less) boundary components. We distinguish them by computing their first Chern classes and three dimensional invariants (whenever possible). Among these contact structures we also distinguish tight ones from those which are overtwisted.Comment: 35 pages, 19 figures, 9 tables (published version

On the support genus of a contact structure

The algorithm given by Akbulut-Ozbagci constructs an explicit open book decomposition on a contact three-manifold described by a contact surgery on a link in the three-sphere. In this article, we will improve this algorithm by using Giroux's contact cell decomposition process. Our algorithm is more economical on choosing the supporting genus of the open book; in particular it gives a good upper bound for the recently defined ``minimal supporting genus invariant'' of contact structures.Comment: 20 pages, 13 figures, title shorthened, minor correction

On Legendrian Embbeddings into Open Book Decompositions

We study Legendrian embeddings of a compact Legendrian submanifold $L$ sitting in a closed contact manifold $(M,\xi)$ whose contact structure is supported by a (contact) open book $\mathcal{OB}$ on $M$. We prove that if $\mathcal{OB}$ has Weinstein pages, then there exist a contact structure $\xi'$ on $M$, isotopic to $\xi$ and supported by $\mathcal{OB}$, and a contactomorphism $f:(M,\xi) \to (M,\xi')$ such that the image $f(L)$ of any such submanifold can be Legendrian isotoped so that it becomes disjoint from the closure of a page of $\mathcal{OB}$.Comment: 14 pages, 4 figures. Major corrections made. arXiv admin note: substantial text overlap with arXiv:1003.220