Log24

Wednesday, February 9, 2011

An Abstract Window

Filed under: General,Geometry — m759 @ 12:00 PM

The sliding window in blue below

http://www.log24.com/log/pix11/110209-SymFrameBWPageSm.jpg

Click for the web page shown.

is an example of a more general concept.

Such a sliding window,* if one-dimensional of length n , can be applied to a sequence of 0's and 1's to yield a sequence of n-dimensional vectors. For example— an "m-sequence" (where the "m" stands for "maximum length") of length 63 can be scanned by a length-6 sliding window to yield all possible 6-dimensional binary vectors except (0,0,0,0,0,0).

For details, see A Galois Field

http://www.log24.com/log/pix11/110209-GaloisStamp.jpg

The image is from Bert Jagers at his page on the Galois field GF(64) that he links to as "A Field of Honor."

For a discussion of the m-sequence shown in circular form above, see Jagers's  "Pseudo-Random Sequences from GF(64)." Here is a noncircular version of the length-63 m-sequence described by Jagers (with length scale below)—

100000100001100010100111101000111001001011011101100110101011111
123456789012345678901234567890123456789012345678901234567890123

This m-sequence may be viewed as a condensed version of 63 of the 64 I Ching  hexagrams. (See related material in this journal.)

For a more literary approach to the window concept, see The Seventh Symbol (scroll down after clicking).

* Moving windows also appear (in a different way) In image processing, as convolution kernels .

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