komm.GolombCode

Golomb code. It is an integer code with domain the positive integers. Let $M \geq 1$ be the divisor. For an integer $n$, let $q$ and $r$ be the quotient and the remainder of the division of $n - 1$ by $M$. The codeword for $n$ consists of the unary codeword of $q + 1$ with stop bit $0$ (that is, $q$ ones followed by a single zero), followed by the truncated binary codeword of $r$ with cardinality $M$ (empty if $M = 1$). For more details, see Gol66 or Say06, Sec. 3.5, where the domain is the non-negative integers, so that the codeword for $n$ here is their codeword for $n - 1$.

The table below shows the codewords for the first integers, with a space between the two parts.

$n$	$M = 1$	$M = 2$	$M = 3$	$M = 4$
$1$	0	0 0	0 0	0 00
$2$	10	0 1	0 10	0 01
$3$	110	10 0	0 11	0 10
$4$	1110	10 1	10 0	0 11
$5$	11110	110 0	10 10	10 00
$6$	111110	110 1	10 11	10 01
$7$	1111110	1110 0	110 0	10 10
$8$	11111110	1110 1	110 10	10 11

Notes

• For $M = 1$ it reduces to the unary code with stop bit $0$.
• For $M$ a power of $2$ it is also known as the Rice code.

Parameters:

• divisor (int) –

  The divisor $M$. Must satisfy $M \geq 1$.

encode_single()

Encodes a single integer into its codeword.

Parameters:

• integer (SupportsIndex) –

  The integer to be encoded. Must be in the domain of the code.

Returns:

• codeword (Word) –

  The codeword of the integer. It is a tuple of bits.

Examples:

>>> code = komm.GolombCode(3)
>>> code.encode_single(4)
(1, 0, 0)

decode_single()

Decodes a single codeword from a bit iterator. This method consumes exactly the bits of one codeword, leaving the iterator at the boundary with the next one; any remaining bits are left untouched.

Parameters:

• bits (Iterator[SupportsIndex]) –

  An iterator of bits starting with a complete codeword. It must be an iterator (as returned by iter), not a general iterable, since advancing it is part of the contract.

Returns:

• integer (int) –

  The decoded integer.

Notes

A ValueError is raised if the iterator is exhausted mid-codeword or if an invalid bit is found.

Examples:

>>> code = komm.GolombCode(3)
>>> bits = iter([1, 0, 0, 0, 0])
>>> code.decode_single(bits)
4
>>> list(bits)  # Iterator is left at codeword boundary
[0, 0]

length()

Returns the codeword length $\ell(n)$ for a given integer $n$ in the domain of the code.

Examples:

>>> code = komm.GolombCode(3)
>>> code.length(4)
3

encode()

Encodes a sequence of integers.

Parameters:

• input (ArrayLike) –

  The sequence of integers to be encoded. Must be a 1D-array of positive integers.

Returns:

• output (Array1D[integer]) –

  The concatenation of the codewords. It is a 1D-array of bits.

Examples:

>>> code = komm.GolombCode(3)
>>> code.encode([4, 1, 3])
array([1, 0, 0, 0, 0, 0, 1, 1])

decode()

Decodes a sequence of bits.

Parameters:

• input (ArrayLike) –

  The sequence of bits to be decoded. Must be a 1D-array of bits. Also, the sequence must be a concatenation of codewords (i.e., the output of the encode method).

Returns:

• output (Array1D[integer]) –

  The sequence of decoded integers. It is a 1D-array of positive integers.

Examples:

>>> code = komm.GolombCode(3)
>>> code.decode([1, 0, 0, 0, 0, 0, 1, 1])
array([4, 1, 3])