Utilies

lift(A::MatElem{T}, type::Symbol=:col) where T <: ResElem

Return the matrix whose residue polynomial elements are converted to circulant matrices over the base field.

lift(A::MatElem{T}, type::Symbol=:col) where T <: CTGroupAlgebra

Return the matrix whose group algebra elements are converted to circulant matrices over the base field.

copy(C::T) where T <: AbstractCode

Returns a copy of the code C.

⊕(A::CTMatrixTypes, B::CTMatrixTypes)
direct_sum(A::CTMatrixTypes, B::CTMatrixTypes)

Return the direct sum of the two matrices A and B.

Hamming_distance(u::T, v::T) where T <: Union{CTMatrixTypes, Vector{S}} where S <: Integer
distance(u::T, v::T) where T <: Union{CTMatrixTypes, Vector{S}} where S <: Integer
dist(u::T, v::T) where T <: Union{CTMatrixTypes, Vector{S}} where S <: Integer

Return the Hamming distance between u and v.

Hamming_weight(v::T) where T <: Union{CTMatrixTypes, Vector{S}} where S <: Integer
weight(v::T) where T <: Union{CTMatrixTypes, Vector{S}} where S <: Integer
wt(v::T) where T <: Union{CTMatrixTypes, Vector{S}} where S <: Integer

Return the Hamming weight of v.

Hermitian_conjugate_matrix(A::CTMatrixTypes)

Return the Hermitian conjugate of the matrix A.

Hermitian_inner_product(u::CTMatrixTypes, v::CTMatrixTypes)

Return the Hermitian inner product of u and v.

are_equivalent_basis(basis::Vector{fq_nmod}, basis2::Vector{fq_nmod})

Return true if basis is a scalar multiple of basis2.

are_symplectic_orthogonal(A::CTMatrixTypes, B::CTMatrixTypes)

Return true if the rows of the matrices A and B are symplectic orthogonal.

dual_basis(E::FqNmodFiniteField, F::FqNmodFiniteField, basis::Vector{fq_nmod})
complementary_basis(E::FqNmodFiniteField, F::FqNmodFiniteField, basis::Vector{fq_nmod})

Return the dual (complentary) basis of basis for the extension E/F.

edge_vertex_incidence_graph(G::SimpleGraph{Int})

Return the edge-vertex incidence graph of G along with the vertex incides of the left and right bipartition.

edge_vertex_incidence_matrix(G::SimpleGraph{Int})

Return the edge-vertex incidence matrix of G along with the vertex incides of the left and right bipartition.

expand_matrix(M::CTMatrixTypes, K::FqNmodFiniteField, β::Vector{fq_nmod})

Return the matrix constructed by expanding the elements of M to the subfield K using the basis β for the base ring of M over K.

extract_bipartition(G::SimpleGraph{Int})

Return two vectors representing the vertex indices of each side of the bipartition.

group_algebra_element_to_circulant_matrix(x::CTGroupAlgebra; type::Symbol=:col)

Return the circulant matrix whose first row or column is the coefficients of x if type is :row or :col, respectively.

is_basis(E::FqNmodFiniteField, F::FqNmodFiniteField, basis::Vector{fq_nmod})

Return true and the dual (complementary) basis if basis is a basis for E/F, otherwise return false, missing.

is_extension(E::FqNmodFiniteField, F::FqNmodFiniteField)

Return true if E/F is a valid field extension.

is_normal_basis(E::FqNmodFiniteField, F::FqNmodFiniteField, basis::Vector{fq_nmod})

Return true if basis is a normal basis for E/F.

is_primitive_basis(E::FqNmodFiniteField, F::FqNmodFiniteField, basis::Vector{fq_nmod})

Return true if basis is a primitive basis for E/F.

is_self_dual_basis(E::FqNmodFiniteField, F::FqNmodFiniteField, basis::Vector{fq_nmod})

Return true if basis is equal to its dual.

is_triorthogonal(G::CTMatrixTypes, verbose::Bool=false)
is_triorthogonal(G::Matrix{Int}, verbose::Bool=false)

Return true if the binary matrix G is triorthogonal.

Notes

• If the optional parameter verbos is set to true, the first pair or triple of non-orthogonal rows will be identified on the console.

is_valid_bipartition(G::SimpleGraph{Int}, left::Vector{Int}, right::Vector{Int})

Return true if the vertices indexed by left and right form a valid bipartition for G.

primitive_basis(E::FqNmodFiniteField, F::FqNmodFiniteField)

Return a primitive basis for E/F and its dual (complementary) basis.

pseudoinverse(M::CTMatrixTypes)

Return the pseudoinverse of a stabilizer matrix M over a quadratic extension.

Notes

• This is not the Penrose-Moore pseudoinverse.

quadratic_residues(q::Int, n::Int)

Return the sets of quadratic resides and quadratic non-residues of q and n.

residue_polynomial_to_circulant_matrix(f::ResElem)

Return the circulant matrix whose first row or column is the coefficients of `f` if `type` is `:row` or `:col`, respectively.

" row_supports(M::CTMatrixTypes)

Returns a vector where the ith entry lists the indices of the nonzero entries of M[i, :]

" rowsupportssymplectic(M::CTMatrixTypes)

Returns a vector where the ith entry is a 2-tuple of lists with the indices of the nonzero X and Z entries of M[i, :]

strongly_lower_triangular_reduction(A::CTMatrixTypes)

Return a strongly lower triangular basis for the kernel of A and a unit vector basis for the complement of the image of transpose(A).

• Note
• This implements Algorithm 1 from https://doi.org/10.48550/arXiv.2204.10812

symplectic_inner_product(u::CTMatrixTypes, v::CTMatrixTypes)

Return the symplectic inner product of u and v.

verify_dual_basis(E::FqNmodFiniteField, F::FqNmodFiniteField, basis::Vector{fq_nmod}, dual_basis::Vector{fq_nmod})
verify_complementary_basis(E::FqNmodFiniteField, F::FqNmodFiniteField, basis::Vector{fq_nmod}, dual_basis::Vector{fq_nmod})

Return true if basis is the dual of dual_basis for E/F, otherwise return false.

wt(f::CTPolyRingElem)

Return the number of nonzero coefficients of the polynomial f.

⊗(A::CTMatrixTypes, B::CTMatrixTypes)
kron(A::CTMatrixTypes, B::CTMatrixTypes)
tensor_product(A::CTMatrixTypes, B::CTMatrixTypes)
kronecker_product(A::CTMatrixTypes, B::CTMatrixTypes)

Return the Kronecker product of the two matrices A and B.

is_regular(G::SimpleGraph{Int})

Return true if G is regular.

tr(x::fq_nmod, K::FqNmodFiniteField, verify::Bool=false)

Return the relative trace of x from its base field to the field K.

Notes

• If the optional parameter verify is set to true, the two fields are checked for compatibility.