Computing parity of a binary word

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 x and y is given by the parity of x&y.

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.

gcc extensions

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

Stand-alone code

If you want your own code for computing parity, Hacker’s Delight gives the following for a 32-bit integer x.

The last bit of y is the parity of x.

    y = x ^ (x >> 1);
    y = y ^ (y >> 2);
    y = y ^ (y >> 4);
    y = y ^ (y >> 8);
    y = y ^ (y >> 16);

More bit twiddling posts

2 thoughts on “Computing parity of a binary word

  1. Something’s wrong with your example from Hacker’s Delight.
    If I put “5” in to what you have written,
    101
    111
    110

    Six isn’t the right answer.
    Compilers have slowly been incorporating a lot of these common algorithms to their optimizers…
    https://godbolt.org/z/eNP4MJ

    Note that “simple_bits” compiles down to the same code as the builtin without relying on non-standard compiler features while the hacker’s version hasn’t made it there. There’s a fun article by the owner of godbolt.org on ACM’s site about some of the craziness under the hood:
    https://queue.acm.org/detail.cfm?id=3372264

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