A quarter turn by this circle acts as multiplication by " i " on the tangent space at the identity coset.
42.
The coset serves as the identity in this group, and the inverse of " gN " in the quotient group is.
43.
Since every right coset is a left coset, there is no need to distinguish " left cosets " from " right cosets ".
44.
Since every right coset is a left coset, there is no need to distinguish " left cosets " from " right cosets ".
45.
Note also that selecting different coset leaders will create a slightly different but equivalent standard array, and will not affect results when decoding.
46.
In the profinite case there are many subgroups of finite index, and Haar measure of a coset will be the reciprocal of the index.
47.
The gluing map ? need only be specified up to taking a double coset in the mapping class group of " H ".
48.
Coset leaders are used in the construction of a standard array for a linear code, which can then be used to decode received vectors.
49.
The Todd-Coxeter algorithm takes a finite index subgroup given by generators of a finitely presented group and gives the multiplication amongst coset representations.
50.
For example, the syndrome-coset uniqueness property of linear block codes is used in trellis shaping, one of the best known shaping codes.
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