Comments by "MC116" (@angelmendez-rivera351) on "The Hidden Geometry of Error-Free Communication" video.

  1. Yes, this is correct. RGB systems are actually operating on 3-dimensional spaces similar to a vector space, and the basis vectors are provided by the encodings of the colors red, blue, and green.
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  2. Consider the set {0, 1}, and define ⊕ : {0, 1}^2 —> {0, 1} by 0 ⊕ X = X ⊕ 0 = X and 1 ⊕ 1 = 0, and define • : {0, 1}^2 —> {0, 1} by 1 • X = X • 1 = X and 0 • 0 = 0. ({0, 1}, ⊕, •) is a field, and it is the Galois field of order 2, also called the Boolean field, which is denoted F(2). Now, let n be a natural number, expresed as a set. Let C(n) be some nonempty set of functions f : n —> {0, 1}, and define + : C(n)^2 —> C(n) such that for all m in n, (f + g)(m) = f(m) ⊕ g(m), and · : {0, 1}×C(n) —> C(n) such that for all m in n, (k·f)(m) = k•f(m). (C(n), +, ·) is a vector space over the field F(2), and this vector space is the definition of a binary linear code. The functions in C(n) are the words of the code, and n is the length of all the words. This vector space can be equipped with the L1 norm, the taxicab norm, with w : C(n) —> [0, ∞), such that w(f) = Σ{|f|}. This norm is called the weight of a word. Thus, binary linear code is defined as Boolean normed space.
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  3. This is not a great suggestion. He already has to cut off explanations about a topic into multiple parts anyway. If every video was sub-30 minutes, then it would take forever to actually get through the topic.
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  4. This is a rather unnatural way to look at it. The natural way to define the weight of a code word is by using the L1 norm.
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  5. Hint: how many vertices does an icosahedron have?
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