Bit-level properties of integers

The integer_bits module defines functions that return bit-level properties of integers, such as partity and bitwise inner product.

AUTHORS:

• Paul Leopardi (2016-08-21): initial version

boolean_cayley_graphs.integer_bits.base2(dim, num)

Map num to \(\mathbb{F}_2^{dim}\) using lexicographical ordering.

INPUT:

• num – non-negative integer. The value to be mapped.

• dim – positive integer. The Boolean dimension.

OUTPUT:

A list of 0,1 integer values of length dim.

EXAMPLES:

sage: from boolean_cayley_graphs.integer_bits import base2
sage: base2(5,3)
[1, 1, 0, 0, 0]
sage: base2(3,5)
[1, 0, 1]
sage: base2(3,1)
[1, 0, 0]

boolean_cayley_graphs.integer_bits.inner(a, b)

Return the inner product of two non-negative integers interpreted as Boolean vectors.

Given the non-negative numbers a and b, the function inner returns the Boolean inner product of their binary expansions.

INPUT:

• a – non-negative integer.

• b – non-negative integer.

OUTPUT:

0 or 1, being the Boolean inner product of the Boolean vectors represented by a and b.

EXAMPLES:

sage: from boolean_cayley_graphs.integer_bits import inner
sage: inner(0,0)
0
sage: inner(1,1)
1
sage: inner(3,3)
0

boolean_cayley_graphs.integer_bits.parity(n)

Return the bit parity of a non-negative integer.

Given the non-negative number n, the function parity returns 1 if the number of 1 bits in the binary expansion is odd, otherwise 0.

INPUT:

• n – non-negative integer.

OUTPUT:

0 or 1, being the bit parity of n.

EXAMPLES:

sage: from boolean_cayley_graphs.integer_bits import parity
sage: parity(0)
0
sage: parity(2)
1
sage: parity(3)
0