TriSym.jl

A Nemo/Julia package to compute trisymmetric decompositions.

Installation

The package can be installed via the package manager Pkg, that is entered with the key ].

(v1.2) pkg> add https://github.com/erou/TriSym.jl

Usage

The package allows to work with Nemo's finite fields but also with abstract commutative algebra over finite fields, i.e. algebras given by their multiplication tensor.

Finite fields

Here is an example with the finite field with 27 elements where we search for decomposition of length 6 with two different margins.

julia> k, x = FiniteField(3, 3, "x")
(Finite field of degree 3 over F_3, x)

julia> multiplication_search(k, (0, 0, 0), 6)
1-element Array{Tuple{Vararg{Tuple{fq_nmod,Int64},N} where N},1}:
 ((x^2+1, 1), (x^2+2*x+1, 2), (2*x^2+x+1, 1), (x^2+x, 2), (2*x^2+x, 2), (x^2, 1))

julia> multiplication_search(k, (2, 1, 0), 6)
4-element Array{Tuple{Vararg{Tuple{fq_nmod,Int64},N} where N},1}:
 ((x+1, 1), (2*x+1, 2), (2*x^2+1, 1), (x, 1), (x^2+x, 2), (2*x^2+x, 1))                    
 ((x^2+1, 1), (x^2+2*x+1, 2), (2*x^2+x+1, 1), (x^2+x, 2), (2*x^2+x, 2), (x^2, 1))          
 ((x^2+1, 2), (x^2+x+1, 1), (2*x^2+1, 2), (2*x^2+2*x+1, 2), (2*x^2+x, 1), (x^2, 2))        
 ((x^2+x+1, 2), (x^2+2*x+1, 1), (2*x^2+1, 1), (2*x^2+x+1, 2), (2*x^2+2*x+1, 1), (x^2+x, 1))

Abstract algebras

Here is the same example as before but in the case of abstract algebras, where the user can specify the multiplication tensor. In this case we give the exact same finite field k, but it also possible to call AbstractCommutativeAlgebra with the matrix of the multiplication tensor.

julia> k
Finite field of degree 3 over F_3

julia> A = TriSym.AbstractCommutativeAlgebra(k)
TriSym.AbstractCommutativeAlgebra{3}([1  0  0  0  2  0]
[0  1  0  0  1  2]
[0  0  1  1  0  1], 3)

julia> L = MatrixSpace(ResidueRing(ZZ, 3), 1, 3)([ResidueRing(ZZ, 3)(coeff(tr(x^i), 0)) for i in 0:3-1])
[0  0  2]

julia> multiplication_search(A, L, (0, 0, 0), 6)
1-element Array{Tuple{Vararg{Tuple{nmod_mat,Int64},N} where N},1}:
 (([1]
[0]
[1], 1), ([1]
[2]
[1], 2), ([1]
[1]
[2], 1), ([0]
[1]
[1], 2), ([0]
[1]
[2], 2), ([0]
[0]
[1], 1))

Tests

In the package manager, the package can be tested via

(v1.2) pkg> test TriSym

Help

In Julia, help can be obtained by pressing the ? key.

help?> multiplication_bilinear_map
search: multiplication_bilinear_map

  multiplication_bilinear_map(k::Nemo.FqNmodFiniteField)

  Compute the bilinear map associated with the multiplication of a finite field k.

  ─────────────────────────────────────────────────────────────────────────────────────────────────────

  multiplication_bilinear_map(A::AbstractCommutativeAlgebra{N}) where N

  Compute the bilinear map assciated with the multiplication in the algebra A.

erou / TriSym.jl