Introduction

A gf2::BitSpan is a non-owning view of bit elements stored compactly in an array of unsigned integer words. Typically bit-spans are created as views into existing gf2::BitSpan or gf2::BitVec objects, but they can also be created from an existing bit-span or even from raw pointers to word arrays.

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

A span is a cheap way to work with a slice of a gf2::BitSpan, gf2::BitVec, or any other gf2::BitStore. It never allocates or copies; it just remembers a pointer into the backing words, a bit-offset, and a length.

Note: Operations on and between bit-spans 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.

Use spans when you need fast, zero-copy views for algorithms (e.g., matrix row slices, polynomial segments, or temporary windows) without giving up the rich gf2::BitStore operations.

Because they are non-owning you can pass spans by value without worrying about copies.

This is similar in spirit to std::span for arrays of regular types, but specialised for bit-level access and manipulation.

Warning: Because a gf2::BitSpan keeps a pointer back to the underlying gf2::BitStore, that store must outlive the span. Outside of that lifetime guarantee, you can freely pass bit spans by value – they are tiny and intentionally lightweight.

Declaration

The declaration of the bit-span class looks like:

template <Unsigned Word>
class BitSpan {
public:
    using word_type = Word;
    // ...
};

In the case of a a bit-span, the const-ness of the Word type determines whether the span is read-only or read-write. A span created using gf2::BitStore::span(begin, end) const will have Word as a const type, while one created using gf2::BitStore::span(begin, end) will have a non-const Word type. The former is a read-only view, while the latter allows mutation.

Note: Spans are lightweight and you generally pass them by value so fun(BitSpan<Word> span) is preferred over fun(BitSpan<Word> const& span). This form also emphasises that what matters is the interior const-ness of the span, not whether the span object itself is const.

Methods Overview

The gf2::BitSpan class provides a rich set of methods for querying and manipulating, bit-spans. 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-spans.
Bit Access	Methods to access individual bit elements in a bit-span.
Queries	Methods to query the overall state of a bit-span.
Mutators	Methods to mutate the overall state of a bit-span.
Fills	Methods to fill a bit-span from various sources.
Spans	Methods to create non-owning views over a part of a bit-span — sib-bit-spans.
Sub-vectors	Methods to pull out a clone of piece of a bit-span as new bit-vector.
Riffling	Methods to create bit-vectors that copy a bit-span with interleaved zeros.
Set/Unset Indices	Methods to find the indices of set & unset bits in a bit-span.
Iterators	Methods to create various iterators over a bit-span.
Stringification	Methods to create string representations of a bit-span.
Equality Operator	Operator to compare bit-stores, including bit-spans for content equality.
Bit Shifts	Operators to shift the bits in bit-spans 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-spans.

Concept Methods

Bit-spans 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::BitSpan::size	Returns the number of bit elements in the span.
gf2::BitSpan::words	Returns the minimum number of Words needed to store those elements.
gf2::BitSpan::word	Returns the "word" for the passed index.
gf2::BitSpan::set_word	Sets the "word" at the passed index to the passed value.
gf2::BitSpan::store const	Returns a const pointer to the beginning of the underlying store.
gf2::BitSpan::store	Returns a non-const pointer to the beginning of the underlying store.
gf2::BitSpan::offset	The bit-span begins at this bit-offset inside its first word.

These methods were trivial to implement for bit sets and vectors but require some thought for bit spans.

The key thing to understand is that all functions operating on a gf2::BitStore operates as if the span is a contiguous array of bits starting at bit-index 0.

Of course, the actual bits in the span may start part-way through a word in the underlying store, so we have to adjust for that in our implementation.

For a bit-span, the return value from the gf2::BitSpan::word(i) method will often be synthesised from two contiguous "real" words w[j] and w[j+1] for some j:

word[i] will use some high-order bits from w[j] and low-order bits from w[j+1] as shown in the following example:

A bit-span with 20 elements, where the `X` is always zero

The BitSpan class behaves as if bits from the real underlying store were copied and shuffled down so that element zero is bit 0 of word 0 in the bit-span. However, it never actually copies anything; instead, it synthesises "words" as needed.

The same principle applies to the gf2::BitSpan::set_word(i, value) method. In the case of a bit-span, calls to set_word(i, value) will generally copy low-order bits from value into the high-order bits of some real underlying word w[j] and copy the rest of the high-order bits from value into the low-order bits of w[j+1].

The other bits in w[j] and w[j+1] will not be touched.

Warning: While the gf2::BitSpan::store method provides write access to the underlying words, this is primarily for internal use. If you do use the pointer, you must ensure that any bits outside the span's range remain unaltered. The set_word method takes care of this for you, so prefer using that method when possible.

Constructors

The gf2::BitSpan class provides just one constructor to create bit-span objects:

BitSpan(Word* data, size_t offset, size_t length);

This constructor creates a bit-span that views length bits starting at bit-offset offset inside the array of words pointed to by data. If the Word type is const, the span will be read-only; otherwise, it will allow mutation.

However, you don't usually construct BitSpan objects directly. Instead, you typically obtain them from existing gf2::BitStore objects using their span member functions:

gf2::BitXXX::span(begin, end) const creates a bit-span as a read-only view of the bits in the half open range [begin, end).
gf2::BitXXX::span(begin, end) creates a bit-span as a read-write view of the bits in the half open range [begin, end).

Here BitXXX can be any class that satisfies the gf2::BitStore concept, such as gf2::BitVec, gf2::BitSet, or even gf2::BitSpan as taking a span of a span is supported.

Those methods use the generic gf2::span method defined for the gf2::BitStore concept as described here documentation.

Spans can start in the middle of a machine word and can end part-way through another word; the class synthesises whole-word reads/writes for you.

Note: Because a gf2::BitSpan keeps a pointer back to the underlying word store, that store must outlive the span. In the Rust version of the library, this is enforced using lifetime parameters. In C++, this is just a convention the user must follow. We could use smart pointers to enforce this at runtime, but that would add overhead and complexity to the concrete gf2::BitStore types.

Example

auto v = gf2::BitVec<u8>::from_string("1111'1111'1111").value();
v.span(2, 6).flip_all();
assert_eq(v.to_string(), "110000111111");

Bit Access

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

Function	Description
gf2::BitSpan::get	Returns the value of a single bit element as a read-only boolean.
gf2::BitSpan::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::BitSpan::front	Returns the value of the first element in the bit-span.
gf2::BitSpan::back	Returns the value of the last element in the bit-span.
gf2::BitSpan::set	Sets a bit to the given boolean value which defaults to true.
gf2::BitSpan::flip	Flips the value of the bit element at a given index.
gf2::BitSpan::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::BitSpan::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:

s[12] = true;

Queries

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

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

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

Mutators

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

Function	Description
gf2::BitSpan::set_all	Sets all the bits in the span to the passed value, which defaults to true.
gf2::BitSpan::flip_all	Flips the values of all the bits in the span.

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

Fills

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

Function	Description
gf2::BitSpan::copy	Makes the bits in this span identical to those from various sources.
gf2::BitSpan::random_fill	Fills the span with random 0's and 1's.

Copies

The gf2::BitSpan::copy methods support copying bits from:

Another bit-store of the same size but possibly a different underlying word type.
A std::bitset of the same size as the bit-span.
An unsigned integer that has the same number of bits as the span. The integer type need not be the same as the underlying Word used by the bit-span.
A function or callable object that takes a single usize index argument and returns a boolean value for that index.

Note: In each case, the size of the source and destinations 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 sub-span 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.

Spans

The following methods let you create a gf2::BitSpan from a bit-span — a sub-bit-span.

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

There are two overloads of the gf2::BitSpan::span method — one for const bit-spans and one for non-const bit-spans:

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

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

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

The gf2::BitSpan::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

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

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

If the bit-span 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-span (there is no trailing zero at the end).

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

Set/Unset Bit Indices

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

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

Iterators

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

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::BitSpan::bits	Returns a gf2::BitsIter iterator over the bits in the bit-span.
gf2::BitSpan::set_bits	Returns a gf2::SetBitsIter iterator to view the indices of all the set bits.
gf2::BitSpan::unset_bits	Returns a gf2::UnsetBitsIter iterator to view the indices of all the unset bits.
gf2::BitSpan::store_words	Returns a gf2::WordsIter iterator to view the "words" underlying the bit-span.
gf2::BitSpan::to_words	Returns a copy of the "words" underlying the bit-span.

There are two overloads of the gf2::BitSpan::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::BitsIter that allows modification of the bits in the bit-span.
Returns an immutable gf2::BitsIter that only allows one to view the bits in the bit-span.

Stringification

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

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

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-spans for content equality.
Bit Shifts	Operators to shift the bits in bit-spans 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-spans.