402 research outputs found
Adaptive Cut Generation Algorithm for Improved Linear Programming Decoding of Binary Linear Codes
Linear programming (LP) decoding approximates maximum-likelihood (ML)
decoding of a linear block code by relaxing the equivalent ML integer
programming (IP) problem into a more easily solved LP problem. The LP problem
is defined by a set of box constraints together with a set of linear
inequalities called "parity inequalities" that are derived from the constraints
represented by the rows of a parity-check matrix of the code and can be added
iteratively and adaptively. In this paper, we first derive a new necessary
condition and a new sufficient condition for a violated parity inequality
constraint, or "cut," at a point in the unit hypercube. Then, we propose a new
and effective algorithm to generate parity inequalities derived from certain
additional redundant parity check (RPC) constraints that can eliminate
pseudocodewords produced by the LP decoder, often significantly improving the
decoder error-rate performance. The cut-generating algorithm is based upon a
specific transformation of an initial parity-check matrix of the linear block
code. We also design two variations of the proposed decoder to make it more
efficient when it is combined with the new cut-generating algorithm. Simulation
results for several low-density parity-check (LDPC) codes demonstrate that the
proposed decoding algorithms significantly narrow the performance gap between
LP decoding and ML decoding
Mathematical Programming Decoding of Binary Linear Codes: Theory and Algorithms
Mathematical programming is a branch of applied mathematics and has recently
been used to derive new decoding approaches, challenging established but often
heuristic algorithms based on iterative message passing. Concepts from
mathematical programming used in the context of decoding include linear,
integer, and nonlinear programming, network flows, notions of duality as well
as matroid and polyhedral theory. This survey article reviews and categorizes
decoding methods based on mathematical programming approaches for binary linear
codes over binary-input memoryless symmetric channels.Comment: 17 pages, submitted to the IEEE Transactions on Information Theory.
Published July 201
On the Saddle-point Solution and the Large-Coalition Asymptotics of Fingerprinting Games
We study a fingerprinting game in which the number of colluders and the
collusion channel are unknown. The encoder embeds fingerprints into a host
sequence and provides the decoder with the capability to trace back pirated
copies to the colluders.
Fingerprinting capacity has recently been derived as the limit value of a
sequence of maximin games with mutual information as their payoff functions.
However, these games generally do not admit saddle-point solutions and are very
hard to solve numerically. Here under the so-called Boneh-Shaw marking
assumption, we reformulate the capacity as the value of a single two-person
zero-sum game, and show that it is achieved by a saddle-point solution.
If the maximal coalition size is k and the fingerprinting alphabet is binary,
we show that capacity decays quadratically with k. Furthermore, we prove
rigorously that the asymptotic capacity is 1/(k^2 2ln2) and we confirm our
earlier conjecture that Tardos' choice of the arcsine distribution
asymptotically maximizes the mutual information payoff function while the
interleaving attack minimizes it. Along with the asymptotic behavior, numerical
solutions to the game for small k are also presented.Comment: submitted to IEEE Trans. on Information Forensics and Securit
Partial permutation decoding for binary linear Hadamard codes
Permutation decoding is a technique which involves finding a subset S, called PD-set, of the permutation automorphism group PAut(C) of a code C in order to assist in decoding. A method to obtain s-PD-sets of size s + 1 for partial permutation decoding for the binary linear Hadamard codes H m of length 2 m , for all m ≥ 4 and 1 < s ≤ (2 m − m − 1)/(1 + m) , is described. Moreover, a recursive construction to obtain s-PD-sets of size s + 1 for H m+1 of length 2 m+1 , from a given s-PD-set of the same size for the Hadamard code of half length H m is also established
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