occasional meanderings in physics' brave new world

Name:
Location: New Zealand

Marni D. Sheppeard

## Friday, November 16, 2007

### M Theory Lesson 124

In coding theory, a Tanner graph is a means of dealing with error correcting codes. Nodes on the graph are of two types: digit and subcode. For a simple linear code, the subcode nodes represent rows of the parity check matrix. Mao et al consider the utility of such graphs and their duality, as a kind of Fourier transform duality. The diagram shows two Tanner graphs: the first a multiplicative graph for the $(7,4)$ Hamming code and the second a convolution graph for the dual code. The $+$ bits denote the single bit parity check. Observe how similar the placing of these bits is to the Time vertices of the Space-Time hexagon. The full Hamming code generator matrix is then a $(7,1)$ matrix, with the $7 \times 7$ circulant spatial part.

I am now reading this article on the triality of a Hamming code VOA. This paper embeds the Hamming code operad into the $D_4$ one so that the $D_4$ triality (which appears in the graviweak sector of Garrett's work as the generation structure) restricts to the triality that permutes the mutually orthogonal conformal vectors for the Hamming code.

In his book, Quantum Fluctuations of Spacetime, L. B. Crowell discusses the Hamming code in the context of a Galois field representation associated to General Relativity. That is, the parallel transport of a spacetime vector in a finite element analysis is expressed in terms of field extensions associated to sets of points.

Doug said...

Hi Kea, The Golay code is associated with the dodecahedron in perhaps a similar manner?

Joe Fields [U-IL-Chicago], Decoding the Golay code by hand
Abstract: We demonstrate a method for encoding and decoding the [24,12,8] extended binary Golay code using a simple apparatus. We also present several generalizations of this construction which admit similar decoding algorithms.
http://www.math.uic.edu/~fields/
DecodingGolayHTML/
introduction.html

November 16, 2007 3:12 PM
Kea said...

Yes, Doug, I believe so. Thanks for the link.

November 16, 2007 3:16 PM