The coordinate alphabet as the discriminant module of m ℤ^ι #
Construction A applied to the zero code produces the lattice L₀ with carrier m ℤ^ι inside
ℚ^ι and form B_m (x, y) = (∑ i, xᵢ yᵢ) / m. Its dual carrier is the full integer coordinate
lattice ℤ^ι, so its discriminant group is ℤ^ι / m ℤ^ι, which reduction modulo m identifies
with the word space (ℤ/m)^ι.
This file exhibits that identification as an isometry of finite bilinear modules
A_{L₀} ≅ (ℤ/m)^ι,
where the right-hand side is the coordinate power of the standard alphabet
TauCeti.FiniteBilinearModule.zmodStandard, whose pairing is xy / m. The class of an integer
vector z goes to its coordinatewise reduction, and the discriminant pairing of two such classes
is (∑ i, zᵢ wᵢ) / m modulo ℤ. When m is even, L₀ is an even lattice and the same map is
an isometry of finite quadratic modules for the half-norm convention: the discriminant quadratic
value of the class of z is (∑ i, zᵢ²) / (2m) modulo ℤ.
Through this identification an additive code over ℤ/m is literally a subgroup of the
discriminant module of an integral lattice, which is the form in which codes feed the
isotropic-subgroup and gluing constructions for lattices.
References #
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Chapter 4, §3, and Chapter 7, §§8–9, for codes in discriminant-glue coordinates.
- W. Ebeling, Lattices and Codes, §§1.3 and 3.3.
- V. V. Nikulin, Integral symmetric bilinear forms and some of their applications, §1.1.
TauCeti.IntegralLattice.rankOne, the formal rank-one analogue adapted here to coordinate powers with the Construction A normalization.
The zero-code lattice and its dual carrier #
The Construction A lattice of the zero code: the lattice L₀ with carrier m ℤ^ι in
ℚ^ι, carrying the normalized form B_m.
Its dual carrier is the full integer coordinate lattice, and its discriminant module is the
coordinate alphabet in which codes over ℤ/m live.
Instances For
The dual carrier of the zero-code lattice is ℤ^ι, the Construction A carrier of the
whole word space.
The dual carrier of the zero-code lattice is the integer coordinate lattice, presented as
a ℤ-linear equivalence with ℤ^ι given by the coordinatewise integer cast.
Instances For
The discriminant group as the word space over ℤ/m #
Reduction modulo m identifies the discriminant group of the zero-code lattice with the
word space (ℤ/m)^ι.
Instances For
The discriminant class of an integer vector goes to its coordinatewise reduction modulo
m.
Reduction of a quotient representative whose underlying dual vector has integer coordinates. This form is independent of the proof that the displayed vector belongs to the dual carrier.
The inverse identification sends a word to the discriminant class of any coordinatewise integer lift.
The discriminant pairing of the zero-code lattice is the normalized dot product of integer
lifts, (∑ i, zᵢ wᵢ) / m modulo ℤ.
The discriminant module of the zero-code lattice is the coordinate alphabet over ℤ/m.
The underlying equivalence is reduction modulo m of integer coordinates.
Instances For
The quadratic refinement for an even modulus #
Over an even modulus the zero-code lattice is even: the code ⊥ is quadratically
isotropic for the coordinate alphabet.
The discriminant quadratic value of the zero-code lattice is the normalized sum of squares
of an integer lift, (∑ i, zᵢ²) / (2m) modulo ℤ.
Over an even modulus the discriminant quadratic module of the zero-code lattice is the
quadratic coordinate alphabet over ℤ/m, in the half-norm convention.
Instances For
Forgetting the quadratic map from the even-modulus isometry recovers the bilinear identification of the discriminant module with the coordinate alphabet.