The `BitArray` Class

Introduction

A gf2::BitArray is a fixed size vector of bit elements stored compactly in an array of unsigned integer words.
The size is set at compile time by a template parameter referred to here as N.

The class satisfies the gf2::BitStore concept, which provides a rich API for manipulating the bits in the bit-array. The free functions defined for that concept are also pulled into the class as member functions. For example, if v is a gf2::BitArray, you can call v.count_ones() to count the number of set bits in the bit-array instead of calling the free function gf2::count_ones(v), though both forms are valid.

In mathematical terms, a bitset is a vector over GF2, the simplest Galois-Field with just two elements usually denoted 0 & 1, as the booleans true & false, or as the bits set & unset. Arithmetic over GF(2) is mod 2, so addition/subtraction becomes the XOR operation while multiplication/division becomes AND.

Note

Operations on and between bitsets and other objects in the gf2 library are implemented using bitwise operations on whole underlying words at a time. These operations are highly optimised in modern CPUs, allowing for fast computation even on large bitsets. It also means we never have to worry about overflows or carries as we would with normal integer arithmetic.

The gf2::BitArray class is a hybrid between a [std::array] and a std::bitset, along with extra mathematical features to facilitate numerical work, and in particular, linear algebra.

It is worth noting that a BitArray prints in vector-order.

For example, a bitset of size four will print as $v_0 v_1 v_2 v_3$ with the elements in increasing index order, so the least significant vector element, $v_0$, comes first on the left. Contrast that to a std::bitset, which always prints in bit-order. The equivalent std::bitset with four elements prints as $b_3 b_2 b_1 b_0$ with the least significant bit $b_0$ printed last on the right.

Of course, for many applications, printing in bit-order makes perfect sense. A bitset of four elements initialised to 0x1 will print 1000. A std::bitset prints the same value as 0001, which will be more natural in some settings.

However, our main aim is numerical work, where vector order is more natural. In particular, bit-order is unnatural for matrices over GF(2). It is too confusing to print a matrix in any order other than the one where the (0,0) element is at the top left, and proceed from there.

Declaration

The declaration of the bit-array class looks like:

template <usize N, Unsigned Word = usize>
class BitArray {
public:
    using word_type = Word;
    // ...
};

The first template parameter N specifies the number of bit elements in the bitset, which are all initialised to zero.

The Word template parameter specifies the underlying gf2::Unsigned integer type used to hold the bit-array's bits, and the default is usually the most efficient type for the target platform. On most modern platforms, that usize will be a 64-bit unsigned integer. The choice of underlying word type is exposed via the public word_type type alias as required by the gf2::BitStore concept.

If your application calls for a vast number of bit-arrays with only a few bits each, you might consider using std::uint8_t as the Word type to save memory.

Note

You might notice that many of the doctests in the library use 8-bit underlying words. The reason is we want to exercise the various functions across word boundaries for modest, easily readable bit-stores.

Methods Overview

The gf2::BitArray class provides a rich set of methods for querying and manipulating, bit-arrays. Here is an overview of the main methods available in the class:

Category	Description
Concept Methods	Methods needed to satisfy the `gf2::BitStore` concept.
Constructors	Methods to create bit-arrays with various initial values..
Factory Constructors	Class methods to create bit-arrays with specific fills.
Bit Access	Methods to access individual bit elements in a bit-array.
Queries	Methods to query the overall state of a bit-array.
Mutators	Methods to mutate the overall state of a bit-array.
Fills	Methods to fill a bit-array from various sources.
Exports	Methods to export the bits in the bit-array to various destinations.
Spans	Methods to create non-owning views over a part of a bit-array --- bit-spans.
Sub-vectors	Methods to pull out a clone of piece of a bit-array as new bit-vector.
Riffling	Methods to create bit-vectors that copy a bit-array with interleaved zeros.
Set/Unset Indices	Methods to find the indices of set & unset bits in a bit-array.
Iterators	Methods to create various iterators over a bit-array.
Stringification	Methods to create string representations of a bit-array.
Equality Operator	Operator to compare bit-stores, including bit-arrays for content equality.
Bit Shifts	Operators to shift the bits in bit-arrays left or right.
Bit-wise Operators	Operators to combine bit-stores using logical operations.
Arithmetic Operators	Operators to add or subtract bit-stores.
Other Functions	Dot products, convolutions, concatenation etc. for bit-arrays.

Concept Methods {#array-concept}

Bit-arrays satisfy the gf2::BitStore concept and forwards many method calls to free functions defined for that concept. The concept requires us to provide the following methods:

Method Name	Description
`gf2::BitArray::size`	Returns `N`, the number of bit elements in the bitset.
`gf2::BitArray::words`	Returns the number of `Word`s needed to store those elements.
`gf2::BitArray::word`	Returns the word for the passed index.
`gf2::BitArray::set_word`	Sets the word at the passed index to the passed value.
`gf2::BitArray::store const`	Returns a const pointer to the beginning of the `Word` store.
`gf2::BitArray::store`	Returns a non-const pointer to the beginning of the `Word` store.
`gf2::BitArray::offset`	For bitsets, this always returns 0.

These methods are trivial to implement for bit-arrays.

The one place where care is needed is in the set_word method, which must ensure that any bits beyond the bitset's size remain set to zero.

Warning

While the gf2::BitArray::store method provides write access to the underlying words, you must be careful when modifying them directly. You must ensure that any bits beyond the bitset's size remain zero. The gf2::BitArray::set_word method takes care of this for you, so prefer using that method when possible.

Constructors {#array-constructors}

The default constructor for a gf2::BitArray<N> creates an a bit-array with N bits, all initialised to zero. You can also create a bit-vector with an initial fill using the following constructor:

Method Name	Description
`gf2::BitArray()`	Constructs a bit-vector with `N` zero elements.
`gf2::BitArray(Word)`	Constructs a bit-array of size `N` by repeatedly copying the bits from the `Word`

Factory Constructors {#array-factory}

There are also many static factory construction functions.

Method Name	Description
`gf2::BitArray::zeros`	Returns a bit-array where all the elements are 0.
`gf2::BitArray::ones`	Returns a bit-array where all the elements are 1.
`gf2::BitArray::constant`	Returns a bit-array where all the elements are whatever is passed as a `value`.
`gf2::BitArray::unit`	Returns a bit-array where all the elements are zero except for a single 1.
`gf2::BitArray::alternating`	Returns a bit-array where all the elements follow the pattern `101010...`
`gf2::BitArray::from`	Returns a `BitArray` filled with bits copied from various sources.
`gf2::BitArray::random`	Returns a `BitArray` with a random fill seeded from entropy.
`gf2::BitArray::seeded_random`	Returns a `BitArray` with a reproducible random fill.
`gf2::BitArray::biased_random`	Returns a random `BitArray` where you set the probability of bits being 1.

The gf2::BitVector::from factory method is overloaded to allow construction from:

Any unsigned integer value, where we copy the bits in the passed value.
The source type need not be the same as the underlying Word type used by the bit-array. The number of bits in the value must match the size of given by the N template parameter. This is enforced at compile time.
An iteration of any unsigned integer values, where we copy all the bits in that iteration.
The source type need not be the same as the underlying Word type used by the bit-vector. The total number of bits in the iteration must match the size of given by the N template parameter. This is enforced at runtime.
A std::bitset where we copy the bits over.
A callable object (a function of some sort) that generates bits on demand when passed an index.

Bit Access {#array-bit-access}

The following methods provide access to individual bit elements in the bit-array.

Function	Description
`gf2::BitArray::get`	Returns the value of a single bit element as a read-only boolean.
`gf2::BitArray::operator[]()`	Returns a `bool` in the const case and a `BitRef` with read-write access to a bit element in the non-const version.
`gf2::BitArray::front`	Returns the value of the first element in the bit-array.
`gf2::BitArray::back`	Returns the value of the last element in the bit-array.
`gf2::BitArray::set`	Sets a bit to the given boolean value which defaults to `true`.
`gf2::BitArray::flip`	Flips the value of the bit element at a given index.
`gf2::BitArray::swap`	Swaps the values of bit elements at locations `i` and `j`.

Note

You can set the DEBUG flag at compile time to enable bounds checks on the index arguments.

The non-const version of gf2::BitArray::operator[]() returns a gf2::BitRef, which is "reference" to an individual bit in the vector. It is automatically converted to a boolean on reads, but it also allows writes, which means you can write natural-looking single-bit assignments:

v[12] = true;

Queries {#array-queries}

The following methods let you query the overall state of a bit-array.

Function	Description
`gf2::BitArray::is_empty`	Returns true if the bit-array is empty
`gf2::BitArray::any`	Returns true if any bit in the bit-array is set.
`gf2::BitArray::all`	Returns true if every bit in the bit-array is set.
`gf2::BitArray::none`	Returns true if no bit in the bit-array is set.
`gf2::BitArray::count_ones`	Returns the number of set bits in the bit-array.
`gf2::BitArray::count_zeros`	Returns the number of unset bits in the bit-array.
`gf2::BitArray::leading_zeros`	Returns the number of leading unset bits in the bit-array.
`gf2::BitArray::trailing_zeros`	Returns the number of trailing unset bits in the bit-array.

These methods efficiently operate on words at a time, so they are inherently parallel.

Mutators {#array-mutators}

The following methods let you mutate the entire bit-array in a single call.

Function	Description
`gf2::BitVector::set_all`	Sets all the bits in the bit-array to the passed value, which defaults to `true`.
`gf2::BitArray::flip_all`	Flips the values of all the bits in the bit-array.

The methods operate on words at a time, so are inherently parallel.

Copies & Fills {#array-fills}

The following methods let you populate the entire bit-array in a single call.

Function	Description
`gf2::BitArray::copy`	Makes the bits in this bit-array identical to those from various sources.
`gf2::BitArray::fill_random`	Fills the bit-array with random 0's and 1's.

Copies

The gf2::BitArray::copy method is overloaded to copy bit values from various sources, where the size of bit-array must match the number of bits in the source:

Another bit-store of the same size but possibly a different underlying word type.
A single unsigned integer value, which need not be the same type as the underlying Word used here.
An iteration of unsigned integer values, which need not be the same type as the underlying Word used here.
A function or callable object that takes a single usize index argument and returns a boolean value for that index.
A std::bitset of the same size as the bit-array.

Note

In each case, the number of bits in the source and destination must match exactly and that condition is always checked unless the NDEBUG flag is set at compile time. You can always use a gf2::BitSpan to copy a subset of bits if needed.

Random Fills

By default, the random fill method uses a random number generator seeded with system entropy, so the results change from run to run. You can set a specific seed to get reproducible fills.

The default probability that a bit is set is 50%, but you can pass a different probability in the range [0.0, 1.0] if desired.

Exports {#array-exports}

The following overloaded method lets you export the bits in the bit-array to various destinations.

Method	Description
`gf2::BitArray::to_words`	Exports the bits in the bit-array as unsigned words.

The ``gf2::BitArray::to_words` function can be passed an output iterator to fill where we assume:

The output iterator points to a location that can accept values of the underlying word type.
There is enough space at the output location to hold all those words.

If gf2::BitVector::to_words is called with no argument it returns a new std::vector of the underlying word type.

Note: The final word in the output may have unused high-order bits that are guaranteed to be set to zero.

Spans {#array-span}

The following methods let you create a gf2::BitSpan, which is a non-owning view of some contiguous subset of bits in the bit-array.

Function	Description
`gf2::BitArray::span`	Returns a `gf2::BitSpan` encompassing the bits in a half-open range `[begin, end)`.

There are two overloads of the gf2::BitArray::span method --- one for const bit-arrays and one for non-const bit-arrays:

auto span(usize begin, usize end);        // <1>
auto span(usize begin, usize end) const;  // <2>

Returns a mutable gf2::BitSpan that allows modification of the bits in the specified range.
Returns an immutable gf2::BitSpan that does not allow modification of the bits in the specified range.

In both cases, the begin and end arguments define a half-open range of bits in the vector.

Mutability/immutability of the returned BitSpan is deep. The span's mutability reflects that of the underlying vector, so if the vector is mutable, so is the span, and vice versa.

This is similar to the C++20 std::span class for regular data collection types.

Note

A gf2::BitSpan also satisfies the gf2::BitStore concept, so you can take a span of a span.

Sub-vectors {#array-sub-vectors}

The following methods create or fill independent bit-vectors with copies of some contiguous subset of the bits in the vector.

Function	Description
`gf2::BitArray::sub`	Returns a new `gf2::BitVector` encompassing the bits in a half-open range `[begin, end)`.
`gf2::BitArray::split_at`	Fills two bit-vectors with the bits in the ranges `[0, at)` and `[at, size())`.

The gf2::BitArray::split_at method can optionally take two pre-existing bit-vectors to fill, thereby avoiding unnecessary allocations in some iterative algorithms that repeatedly use this method.

Note

These methods do not alter the underlying vector.

Riffling {#array-riffling}

We have methods that can interleave (riffle) the bits in a bit-array with zeros.

Function	Description
`gf2::BitArray::riffled`	Fills a pre-existing bit-vector with the result of riffling this bit-array.
`gf2::BitArray::riffled`	Returns a new bit-vector that is this bit-array with its bits interleaved with zeros.

If the bit-array looks like $v_0 v_1 v_2 \ldots v_n$, then the riffling operation produces the vector $v_0 0 v_1 0 v_2 0 \ldots v_n$ where a zero is interleaved between every bit in the original bit-array (there is no trailing zero at the end).

If you think of a bit-array as representing the coefficients of a polynomial over GF(2), then riffling corresponds to squaring that polynomial. See the documentation for gf2::BitPolynomial::squared for more information.

Set/Unset Bit Indices {#array-indices}

The following methods find the indices of set or unset bits in the vector.

Function	Description
`gf2::BitArray::first_set`	Returns the index of the first set bit in the bit-array.
`gf2::BitArray::last_set`	Returns the index of the last set bit in the bit-array.
`gf2::BitArray::next_set`	Returns the index of the next set bit in the bit-array after the passed index.
`gf2::BitArray::previous_set`	Returns the index of the previous set bit in the bit-array before the passed index.
`gf2::BitArray::first_unset`	Returns the index of the first unset bit in the bit-array.
`gf2::BitArray::last_unset`	Returns the index of the last unset bit in the bit-array.
`gf2::BitArray::next_unset`	Returns the index of the next unset bit in the bit-array after the passed index.
`gf2::BitArray::previous_unset`	Returns the index of the previous unset bit in the bit-array before the passed index.

Iterators {#array-iterators}

The following methods create iterators for traversing the bits or underlying words in the bit-array:

Read-only iteration through the individual bits.
Read-write iteration through the individual bits.
Read-only iteration through the indices of the set bits.
Read-only iteration through the indices of the unset bits.
Read-write iteration through the underlying vector words.

Function	Description
`gf2::BitArray::bits`	Returns a `gf2::Bits` iterator over the bits in the bit-array.
`gf2::BitArray::set_bits`	Returns a `gf2::SetBits` iterator to view the indices of all the set bits.
`gf2::BitArray::unset_bits`	Returns a `gf2::UnsetBits` iterator to view the indices of all the unset bits.
`gf2::BitArray::store_words`	Returns a `gf2::Words` iterator to view the "words" underlying the bit-array.
`gf2::BitArray::to_words`	Returns a copy of the "words" underlying the bit-array.

There are two overloads of the gf2::BitArray::bits method --- one for const bit-stores and one for non-const bit-stores:

auto bits();        // <1>
auto bits() const;  // <2>

Returns a mutable gf2::Bits that allows modification of the bits in the bit-array.
Returns an immutable gf2::Bits that only allows one to view the bits in the bit-array.

Stringification {#array-stringification}

The following methods return a string representation for a bit-array. The string can be in the obvious binary format or a more compact hex format.

Function	Description
`gf2::BitArray::to_string`	Returns a default string representation for a bit-array.
`gf2::BitArray::to_pretty_string`	Returns a "pretty" string representation for a bit-array.
`gf2::BitArray::to_binary_string`	Returns a binary string representation for a bit-array.
`gf2::BitArray::to_hex_string`	Returns a compact hex string representation for a bit-array.

Other Operators and Functions

There are many operators and free functions defined for any gf2::BitStore compatible class, including:

Category	Description
Equality Operator	Operator to compare bit-stores, including bit-arrays for content equality.
Bit Shifts	Operators to shift the bits in bit-arrays left or right.
Bit-wise Operators	Operators to combine bit-stores using logical operations.
Arithmetic Operators	Operators to add or subtract bit-stores.
Other Functions	Dot products, convolutions, concatenation etc. for bit-arrays.

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The `BitArray` Class

Introduction

Declaration

Methods Overview

Concept Methods {#array-concept}

Constructors {#array-constructors}

Factory Constructors {#array-factory}

Bit Access {#array-bit-access}

Queries {#array-queries}

Mutators {#array-mutators}

Copies & Fills {#array-fills}

Copies

Random Fills

Exports {#array-exports}

Spans {#array-span}

Sub-vectors {#array-sub-vectors}

Riffling {#array-riffling}

Set/Unset Bit Indices {#array-indices}

Iterators {#array-iterators}

Stringification {#array-stringification}

Other Operators and Functions

See Also

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The BitArray Class

Introduction

Declaration

Methods Overview

Concept Methods {#array-concept}

Constructors {#array-constructors}

Factory Constructors {#array-factory}

Bit Access {#array-bit-access}

Queries {#array-queries}

Mutators {#array-mutators}

Copies & Fills {#array-fills}

Copies

Random Fills

Exports {#array-exports}

Spans {#array-span}

Sub-vectors {#array-sub-vectors}

Riffling {#array-riffling}

Set/Unset Bit Indices {#array-indices}

Iterators {#array-iterators}

Stringification {#array-stringification}

Other Operators and Functions

See Also

The `BitArray` Class