The previous post mentioned adding a parity bit to a string of bits as a way of detecting errors. The parity of a binary word is 1 if the word contains an odd number of 1s and 0 if it contains an even number of ones.
Codes like the Hamming codes in the previous post can have multiple parity bits. In addition to the parity of a word, you might want to also look at the parity of a bit mask AND’d with the word. For example, the Hamming(7, 4) code presented at the end of that post has three parity bits. For a four-bit integer
x, the first parity bit is the parity bit of
x & 0b1011
the second is the parity bit of
x & 0b1101
and the third is the parity of
x & 0b1110
These three bit masks are the last three columns of the generator matrix for of the Hamming(7, 4) code:
1 0 0 0 1 1 1 0 1 0 0 0 1 1 0 0 1 0 1 0 1 0 0 0 1 1 1 0
More generally, any linear operation on a vector of bits is given by multiplication by a binary matrix, where arithmetic is carried out mod 2. Matrix products can be defined in terms of inner products, and the inner product of words
y is given by the parity of
Given that parity is important to compute, how would you compute it?
If you have a popcount function, you could read the last bit of the popcount. Since popcount counts the number of ones, the parity of
x is 1 if
popcount(x) is odd and 0 otherwise.
In the earlier post on popcount, we mentioned three functions that gcc provides to compute popcount:
__builtin_popcount __builtin_popcountl __builtin_popcountll
These three functions return popcount for an
unsigned int, an
unsigned long, and an
unsigned long long respectively.
In each case the function will call a function provided by the target processor if it is available, and will run its own code otherwise.
There are three extensions for computing parity that are completely analogous:
__builtin_parity __builtin_parityl __builtin_parityll
If you want your own code for computing parity, Hacker’s Delight gives the following for a 32-bit integer
The last bit of
y is the parity of
y = x ^ (x >> 1); y = y ^ (y >> 2); y = y ^ (y >> 4); y = y ^ (y >> 8); y = y ^ (y >> 16);