M Theory Lesson 81

Any linear code over a finite field with a circulant generator matrix, such as the Hamming code, is called a cyclic code. For example, the weight 2 code words (1,1,0), (0,1,1) and (1,0,1) form a cyclic binary code along with the zero word (0,0,0). This code corresponds to an ideal in the field

$\frac{\mathbb{F}_2 [x]}{(x^3 – 1)}$

which is a finite version of the rational field of Lesson 38, which looked at simple rational $3 \times 3$ circulants.

Recall that Carl Brannen’s circulants are instead of multiplicative type since the conjugate off-diagonal elements multiply, as complex numbers, to give 1. This suggests a setting of exponentiated linear circulants, and hence an infinite series representation for the complex elements.

4 Responses so far »

  1. 1

    Matti Pitkanen said,

    Dear Kea,

    the notion of rig allowing to represent algebraic numbers resulting as roots of polynomials with positive natural numbers as coefficients generalizes by replacing natural numbers by p-adic integers: this means that one allows also “super-naturals” identified as p-adics infinite as real numbers.

    One obtains all complex algebraic numbers as cardinalities and in some cases the algebraic functions involved converge also for some p-adic number fields so that set theoretic representations of the object as as p-adic fractal is possible as discussed in the first posting. In particular, the Golden Object of John Baez exists 2-adically.

    See the posting at my blog and its predecessor.

  2. 2

    CarlBrannen said,

    Yes, the exponential maps between Lie groups and Lie algebras. I’m writing up the blog post on Lie groups right now.

    As an aside, possibly related, the quarks of su(3) are usually written as (1,0), (1/2,+sqrt(3)/2), (1/2,-sqrt(3)/2). This is nice in that it uses only 2 degrees of freedom, that is, the quantum numbers for color.

    But a more elegant version of su(3) is to use (1,0,0), (0,1,0), and (0,0,1) where these mean (red,green,blue) and eliminate the extra dimension by letting red+green+blue = 0.

    And an alternative way of doing the same thing is to use (+1,-1,0), (-1,0,+1), (0,+1,-1). This is very similar to the weight code words you’re discussing.

    These are the quantum numbers for the 3 colors. The quantum numbers for the white state is of course (0,0,0).

  3. 3

    Kea said,

    Thanks guys. This is all cool!

  4. 4

    Matti Pitkanen said,

    A little Oops to the previous comment. Golden Object exists not only for p=2 but for every prime satisfying p mod 5 =square since in this case 5 is square mod p by Quadratic Resiprocity. For instance, p=11,19,29,31 are Golden primes.

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