Since the goal of this is to allow fast parallel unpacking, blocks of trits can't be infinitely big. A small "block" size needs to be found, ideally one which is both efficient with information density and which is convenient on current hardware.

To find a good block size, we'll need to find a power of 3 for which the next power of 2 is very close.

It's very fortunate that 5 trits fit quite tight into 8 bits at 1.6 bits per trit. When compared to perfect packing, this is 99.06% efficient.

The basic idea with this packing scheme is simply to make a number out of the ternary digits.

Packing trits into bytes should be similar enough.

While repeated remainder and divisions can be used to extract the digits of a number, the problem with divisions and modulo is that they are not usually supported on integers in SIMD programming.

A way around this is obviously to view numbers differently.

Would it be nice if instead of extracting the least significant digit with modulo, we could extract the most significant digit with a multiplication?

Fixed point numbers to the rescue!

Tada!

Now digits can be easily extracted from the top two bits of the resulting 10-bit number when multiplying this 8-bit byte by 3.

This is much more convenient than modulo when unpacking with SIMD.

The only place where there are divisions in this scheme when packing trits into bytes. This assumes that packing is done less often than unpacking, which is very true in the context of LLM weights.

The relevant interesting line is this one:

It does what is depicted in the diagram above, but multiplication is done first because these are integer operations. Doing a ceiling division here is necessary to cancel the off-by-one error from truncating when extracting digits later.

To unpack without using the modulo operator:

To convince myself that this works, I wrote a C program checking that this really is lossless:

Compile and run with:

And I'm getting PASS for each of the 243 ternary numbers which fit in 8 bits.