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quasi-cyclic_code.jl
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quasi-cyclic_code.jl
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# Copyright (c) 2022, 2023 Eric Sabo
# All rights reserved.
#
# This source code is licensed under the BSD-style license found in the
# LICENSE file in the root directory of this source tree.
#############################
# constructors
#############################
"""
QuasiCyclicCode(A::MatElem{T}, parity::Bool=false) where T <: ResElem
Return the quasi-cycle code specified by the matrix `A` of polynomial circulant generators. If the
optional paramater `parity` is set to `true`, the input is used to construct the parity-check matrix.
"""
function QuasiCyclicCode(A::MatElem{T}, parity::Bool=false) where T <: ResElem
R = parent(A[1, 1])
S = base_ring(R)
F = base_ring(S)
g = modulus(R)
m = degree(g)
g == gen(S)^m - 1 || throw(ArgumentError("Residue ring not of the form x^m - 1."))
l = ncols(A)
if parity
A_type = :H
H = lift(A)
k, _ = right_kernel(H)
W = weight_matrix(A)
return QuasiCyclicCode(F, R, ncols(H), k, missing, 1, ncols(H), missing, missing,
missing, missing, missing, missing, l, m, A, A_type, W, maximum(W))
else
A_type = :G
G = lift(A)
k = rank(G)
W = weight_matrix(A)
return QuasiCyclicCode(F, R, ncols(G), k, missing, 1, ncols(G), missing, missing,
missing, missing, missing, missing, l, m, A, A_type, W, maximum(W))
end
end
"""
QuasiCyclicCode(v::Vector{fq_nmod_mat}, l::Int, circ_gens::Bool, parity::Bool=false)
Return the quasi-cyclic code of index `l` generated by right-bit shifts of size `l` of the
generator vectors `v`. If `circ_gens` is `true`, the vectors are taken to be (column) generators
for the circulant matrices instead of generator vectors for the code. If the optional paramater
`parity` is set to `true`, the input is used to construct the parity-check matrix.
# Notes
* If `circ_gens` is `false`, then the length of the code is `ncols(v[1])` and must be divisible by `l`.
* If `circ_gens` is `true`, then the length of the code is `ncols(v[1]) * l`. Circulant matrices are
stacked in rows of length `l`, so `l` must divide `length(v)`.
"""
function QuasiCyclicCode(v::Vector{T}, l::Int, circ_gens::Bool, parity::Bool=false) where T <: CTMatrixTypes
F = base_ring(v[1])
len_v = length(v)
if circ_gens
len_v >= 2 || throw(ArgumentError("Length of input vector must be at least two."))
len_v % l == 0 || throw(ArgumentError("The length of the input vector must be divisible by l."))
nr = div(len_v, l)
r, m = size(v[1])
(r != 1 && m != 1) && throw(ArgumentError("The input matrices must be vectors."))
m == 1 && (v[1] = transpose(v[1]); (r, m = size(v[1]));)
for i in 2:len_v
F == base_ring(v[i]) || throw(ArgumentError("All inputs must be over the same base ring."))
r2, m2 = size(v[i])
(r2 != 1 && m2 != 1) && throw(ArgumentError("The input matrices must be vectors."))
m2 == 1 && (v[i] = transpose(v[i]); (r2, m2 = size(v[i]));)
m == m2 || throw(ArgumentError("The input vectors must all be the same length."))
end
S, x = PolynomialRing(F, :x)
R = residue_ring(S, x^m - 1)
A = zero_matrix(R, nr, l)
for r in 1:nr
for c in 1:l
temp = [v[(r - 1) * l + c][i] for i in 1:m]
A[r, c] = R(S(temp))
end
end
return QuasiCyclicCode(A, parity)
else
r, n = size(v[1])
(r != 1 && n != 1) && throw(ArgumentError("The input matrices must be vectors."))
n == 1 && (v[1] = transpose(v[1]); (r, n = size(v[1]));)
n % l == 0 || throw(ArgumentError("Parameter l must divide the length of the vector."))
m = div(n, l)
for i in 2:len_v
F == base_ring(v[i]) || throw(ArgumentError("All vectors must be over the same base ring."))
r2, n2 = size(v[i])
(r2 != 1 && n2 != 1) && throw(ArgumentError("The input matrices must be vectors."))
n2 == 1 && (v[i] = transpose(v[i]); (r2, n2 = size(v[i]));)
n == n2 || throw(ArgumentError("The input vectors must all be the same length."))
end
S, x = PolynomialRing(F, :x)
R = residue_ring(S, x^m - 1)
A = zero_matrix(R, len_v, l)
for k in 1:len_v
for i in 1:l
row = v[k]
top_circ_row = zero_matrix(F, m, 1)
for j in 1:m
top_circ_row[j, 1] = row[i + (j - 1) * l]
end
# transpose to get first circulant column
top_circ_row[2:end, :] = top_circ_row[end:-1:2, :]
A[k, i] = R(S([top_circ_row[i, 1] for i in 1:m]))
end
end
return QuasiCyclicCode(A, parity)
end
end
"""
QuasiCyclicCode(v::fq_nmod_mat, l::Int, parity::Bool=false)
Return the quasi-cyclic code of index `l` generated by right-bit shifts of size `l` of the
generator vector `v`. If the optional paramater `parity` is set to `true`, the input is used
to construct the parity check matrix.
"""
QuasiCyclicCode(v::CTMatrixTypes, l::Int, parity::Bool=false) = QuasiCyclicCode([v], l, false, parity)
"""
QuasiCyclicCode(v::Vector{fq_nmod_poly}, n::Int, l::Int, parity::Bool=false)
Return the quasi-cyclic code of index `l` whose circulants are defined by the generator
polynomials `v`. If the optional paramater `parity` is set to `true`, the input is used
to construct the parity check matrix.
"""
function QuasiCyclicCode(v::Vector{T}, n::Int, l::Int, parity::Bool=false) where T <: CTMatrixTypes
# if g = x^10 + α^2*x^9 + x^8 + α*x^7 + x^3 + α^2*x^2 + x + α
# g.coeffs = [α 1 α^2 1 0 0 0 α 1 α^2 1]
gen_vecs = Vector{T}()
F = base_ring(v[1])
for g in v
temp = zero_matrix(F, 1, n)
coeffs = collect(coefficients(g))
temp[1, 1:length(coeffs)] = coeffs
push!(gen_vecs, temp)
end
return QuasiCyclicCode(gen_vecs, l, true, parity)
end
"""
QuasiCyclicCode(v::Vector{AbstractCyclicCode}, l::Int, parity::Bool=false)
Return the quasi-cyclic code of index `l` whose circulants are determined by the cyclic
codes in `v`. If the optional paramater `parity` is set to `true`, the input is used to
construct the parity check matrix.
"""
function QuasiCyclicCode(v::Vector{AbstractCyclicCode}, l::Int, parity::Bool=false)
gen_vecs = Vector{fq_nmod_mat}()
for C in v
push!(gen_vecs, C.G[1, :])
end
return QuasiCyclicCode(gen_vecs, l, true, parity)
end
#############################
# getter functions
#############################
"""
index(C::AbstractQuasiCyclicCode)
Return the index of the quasi-cyclic code.
"""
index(C::AbstractQuasiCyclicCode) = C.l
"""
expansion_factor(C::AbstractQuasiCyclicCode)
Return the expansion factor of the quasi-cycle code `C`.
"""
expansion_factor(C::AbstractQuasiCyclicCode) = C.m
"""
type(C::AbstractQuasiCyclicCode)
Return the type of the quasi-cycle code `C`.
"""
type(C::AbstractQuasiCyclicCode) = C.type
"""
polynomial_matrix(C::AbstractQuasiCyclicCode)
Return the polynomial matrix used to define the code.
Use `polynomial_matrix_type` to determine if specifies the generator or parity-check matrix.
"""
polynomial_matrix(C::AbstractQuasiCyclicCode) = C.A
"""
polynomial_matrix_type(C::AbstractQuasiCyclicCode)
Return `'G'` if the polynomial matrix of `C` specifies the generator or parity-check matrix.
"""
polynomial_matrix_type(C::AbstractQuasiCyclicCode) = C.A_type
"""
is_single_generator(C::AbstractQuasiCyclicCode)
Return `true` if `C` is a single-generator quasi-cyclic code.
"""
is_single_generator(C::AbstractQuasiCyclicCode) = (nrows(C.A) == 1;)
#############################
# setter functions
#############################
#############################
# general functions
#############################
"""
base_matrix(A::MatElem{T}) where T <: ResElem
protograph_matrix(A::MatElem{T}) where T <: ResElem
weight_matrix(A::MatElem{T}) where T <: ResElem
Return the base/protograph/weight matrix of `A`.
"""
function weight_matrix(A::MatElem{T}) where T <: ResElem
nr, nc = size(A)
W = zeros(Int, nr, nc)
for c in 1:nc
for r in 1:nr
W[r, c] = wt(Nemo.lift(A[r, c]))
end
end
return W
end
base_matrix(A::MatElem{T}) where T <: ResElem = weight_matrix(A)
protograph_matrix(A::MatElem{T}) where T <: ResElem = weight_matrix(A)
"""
noncirculant_generator_matrix(C::AbstractQuasiCyclicCode)
Return the non-circulant form of the generator matrix for the quasi-cyclic code `C` if the
polynomial matrix specifies the generator matrix; otherwise, return `missing`.
"""
function noncirculant_generator_matrix(C::AbstractQuasiCyclicCode)
if C.A_type == :G
flag = true
for r in 1:C.m
G_inner = zero_matrix(C.F, C.l, C.l * C.n)
for col in 1:C.l
C = lift(C.A[r, col])
G_inner = zero(C)
for i in 1:m
c = k % l
c == 0 && (c = l;)
G_inner[:, (i - 1) * l + c] = C[:, i]
end
end
flag ? (G = G_inner;) : (G = vcat(G, G_inner);)
end
return G
else
return missing
end
end
"""
noncirculant_parity_check_matrix(C::AbstractQuasiCyclicCode)
Return the non-circulant form of the parity-check matrix for the quasi-cyclic code `C`
if the polynomial matrix specifies the parity-check matrix; otherwise, return `missing`.
"""
function noncirculant_parity_check_matrix(C::AbstractQuasiCyclicCode)
if C.A_type == :H
flag = true
for r in 1:C.m
H_inner = zero_matrix(C.F, C.l, C.l * C.n)
for col in 1:C.l
C = lift(C.A[r, col])
H_inner = zero(C)
for i in 1:m
c = k % l
c == 0 && (c = l;)
H_inner[:, (i - 1) * l + c] = C[:, i]
end
end
flag ? (H = H_inner;) : (H = vcat(H, H_inner);)
end
return H
else
return missing
end
end
"""
generators(C::AbstractQuasiCyclicCode)
Return the generators of the quasi-cyclic code.
"""
function generators(C::AbstractQuasiCyclicCode)
G = noncirculant_generator_matrix(C)
gen_vecs = Vector{fq_nmod_mat}()
nr = nrows(G)
for i in 1:nr
if i % l == 1
push!(gen_vecs, G[i, :])
end
end
return gen_vecs
end
"""
circulants(C::AbstractQuasiCyclicCode)
Return the circulant matrices of the quasi-cyclic code.
"""
function circulants(C::AbstractQuasiCyclicCode)
circulants = Vector{fq_nmod_mat}()
nr, nc = size(C.A)
# want stored in row order
for r in 1:nr
for c in 1:nc
push!(circulants, lift(C.A[r, c]))
end
end
return circulants
end