TY - GEN

T1 - Oblivious channels

AU - Langberg, Michael

PY - 2006

Y1 - 2006

N2 - Let C = {x1, . . ., xN} ⊂ {0, 1}n be an [n,N] binary error correcting code (not necessarily linear). Let e ∈ {0, 1}n be an error vector. A codeword x ∈ C is said to be disturbed by the error e if the closest codeword to x ⊕ e is no longer x. Let Ae be the subset of codewords in C that are disturbed by e. In this work we study the size of Ae in random codes C (i.e. codes in which each codeword xi is chosen uniformly and independently at random from {0, 1}n). Using recent results of Vu [Random Structures and Algorithms 20(3)] on the concentration of non-Lipschitz functions, we show that |Ae| is strongly concentrated for a wide range of values of N and ||e||. We apply this result in the study of communication channels we refer to as oblivious. Roughly speaking, a channel W(y|x) is said to be oblivious if the error distribution imposed by the channel is independent of the transmitted codeword x. For example, the well studied Binary Symmetric Channel is an oblivious channel. In this work, we define oblivious and partially oblivious channels and present lower bounds on their capacity. The oblivious channels we define have connections to Arbitrarily Varying Channels with state constraints.

AB - Let C = {x1, . . ., xN} ⊂ {0, 1}n be an [n,N] binary error correcting code (not necessarily linear). Let e ∈ {0, 1}n be an error vector. A codeword x ∈ C is said to be disturbed by the error e if the closest codeword to x ⊕ e is no longer x. Let Ae be the subset of codewords in C that are disturbed by e. In this work we study the size of Ae in random codes C (i.e. codes in which each codeword xi is chosen uniformly and independently at random from {0, 1}n). Using recent results of Vu [Random Structures and Algorithms 20(3)] on the concentration of non-Lipschitz functions, we show that |Ae| is strongly concentrated for a wide range of values of N and ||e||. We apply this result in the study of communication channels we refer to as oblivious. Roughly speaking, a channel W(y|x) is said to be oblivious if the error distribution imposed by the channel is independent of the transmitted codeword x. For example, the well studied Binary Symmetric Channel is an oblivious channel. In this work, we define oblivious and partially oblivious channels and present lower bounds on their capacity. The oblivious channels we define have connections to Arbitrarily Varying Channels with state constraints.

UR - http://www.scopus.com/inward/record.url?scp=39049171461&partnerID=8YFLogxK

U2 - 10.1109/ISIT.2006.261560

DO - 10.1109/ISIT.2006.261560

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AN - SCOPUS:39049171461

SN - 1424405041

SN - 9781424405046

T3 - IEEE International Symposium on Information Theory - Proceedings

SP - 2739

EP - 2743

BT - Proceedings - 2006 IEEE International Symposium on Information Theory, ISIT 2006

T2 - 2006 IEEE International Symposium on Information Theory, ISIT 2006

Y2 - 9 July 2006 through 14 July 2006

ER -