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, Hackerâ€™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);

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

Thanks. I had left out an important sentence that I just added: The last bit of

`y`

is the parity of`x`

.