This package provides types and functions for linear algebra modulo 2. It defines a type ZZ2 for integers mod 2 and ZZ2Array (ZZ2Vector, ZZ2Matrix) for arrays (vectors, matrices) with elements in ZZ2.

The type ZZ2 <: Number represents integers modulo 2. It is simply a wrapper around Bool that allows to use the usual arithmetic operations + and * instead of the logical operations xor and &.

Elements can be created from Bool or any other Integer type or via the functions zero and one. Similarly, Integer types are promoted to ZZ2 when needed.

julia> ZZ2(1) == one(ZZ2)
true

julia> iszero(ZZ2(4))
true

julia> ZZ2(1) + 3
0

julia> typeof(ans)
ZZ2

The type ZZ2Array{N} <: AbstractArray{ZZ2,N} represents N-dimensional arrays with elements of type ZZ2. There are the usual abbreviations ZZ2Vector for ZZ2Array{1} and ZZ2Matrix for ZZ2Array{2}.

The internal representation is packed, meaning that each element only uses one bit. However, columns are internally padded to a length that is a multiple of 256.

A ZZ2Array can be created from any AbstractArray whose elements can be converted to ZZ2. One can also leave the elements undefined by using the undef argument.

julia> ZZ2Matrix([1 2 3; 4 5 6])
2×3 ZZ2Matrix:
 1  0  1
 0  1  0

julia> v = ZZ2Vector(undef, 2); v[1] = true; v[2] = 2.0; v
2-element ZZ2Vector:
 1
 0

Broadcasting is supported to some extent:

julia> a = ZZ2Matrix([1 0 1; 1 1 0])
2×3 ZZ2Matrix:
 1  0  1
 1  1  0

julia> b = ones(ZZ2, 2, 3)
2×3 ZZ2Matrix:
 1  1  1
 1  1  1

julia> a .+= b
2×3 ZZ2Matrix:
 0  1  0
 0  0  1

The package defines functions for (reduced) row and column echelon forms (rref, rcef), rank, determinant (det) and inverses (inv). There are also the argument-modifying counterparts rcef!, rank!, det! and inv!. See the docstrings.

Below we compare Modulo2.jl to the Julia packages LinearAlgebraX.jl (with Mods.jl for integers mod 2), AbstractAlgebra.jl and Nemo.jl. Note that unlike Modulo2.jl these packages can deal with more general coefficients than integers mod 2. Also keep in mind that the precise results depend on the specific processor and memory.

The timings are for square matrices of the size indicated.

Matrix{ZZ2} refers to Julia's standard matrix multiplication with coefficients in ZZ2.

Package versions: Mods v2.2.4, LinearAlgebraX v0.2.7, AbstractAlgebra v0.40.1, Nemo v0.43.1

Computer: Intel Core i3-10110U CPU @ 2.10GHz with 8GB RAM

The benchmark code can be found in the benchmark directory.