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Publication# Conservation laws for coding

Abstract

This work deals with coding systems based on sparse graph codes. The key issue we address is the relationship between iterative (in particular belief propagation) and maximum a posteriori decoding. We show that between the two there is a fundamental connection, which is reminiscent of the Maxwell construction in thermodynamics. The main objects we consider are EXIT-like functions. EXIT functions were originally introduced as handy tools for the design of iterative coding systems. It gradually became clear that EXIT functions possess several fundamental properties. Many of these properties, however, apply only to the erasure case. This motivates us to introduce GEXIT functions that coincide with EXIT functions over the erasure channel. In many aspects, GEXIT functions over general memoryless output-symmetric channels play the same role as EXIT functions do over the erasure channel. In particular, GEXIT functions are characterized by the general area theorem. As a first consequence, we demonstrate that in order for the rate of an ensemble of codes to approach the capacity under belief propagation decoding, the GEXIT functions of the component codes have to be matched perfectly. This statement was previously known as the matching condition for the erasure case. We then use these GEXIT functions to show that in the limit of large blocklengths a fundamental connection appears between belief propagation and maximum a posteriori decoding. A decoding algorithm, which we call Maxwell decoder, provides an operational interpretation of this relationship for the erasure case. Both the algorithm and the analysis of the decoder are the translation of the Maxwell construction from statistical mechanics to the context of probabilistic decoding. We take the first steps to extend this construction to general memoryless output-symmetric channels. More exactly, a general upper bound on the maximum a posteriori threshold for sparse graph codes is given. It is conjectured that the fundamental connection between belief propagation and maximum a posteriori decoding carries over to the general case.

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Pythagorean theorem

In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides. The theorem can be written as an equation relating the lengths of the sides a, b and the hypotenuse c, sometimes called the Pythagorean equation: The theorem is named for the Greek philosopher Pythagoras, born around 570 BC.

Binary erasure channel

In coding theory and information theory, a binary erasure channel (BEC) is a communications channel model. A transmitter sends a bit (a zero or a one), and the receiver either receives the bit correctly, or with some probability receives a message that the bit was not received ("erased") . A binary erasure channel with erasure probability is a channel with binary input, ternary output, and probability of erasure . That is, let be the transmitted random variable with alphabet .