Generate a unique 2-byte integer from three integers

I have three integers:

First number ranges from 1-600
Second number ranges from 0-4
Third number ranges from 0-14

I need to generate a 2-byte unique integer from the above integers such that:

Using the same number again would result in the same number
Ability to get back all three numbers from the unique number
Unique number should be > 0 && < 32768 (2-byte)

I have searched for solutions, but most solution doubles the number of bytes from 2 to 4 bytes which is not an option in my case.

Solution

First, let's convert the first range to 0-599, marked as first-1 below. Then, all you need to do is multiply the numbers together, using range maximums as multipliers:

range_of_first = 599 + 1 = 600
range_of_second = 4 + 1 = 5
range_of_third = 14 + 1 = 15

(first-1) + range_of_first * second + range_of_first * range_of_second * third
= (first-1) + 600*second + 600*5*third
= result

Maximum value you can get is 599 + 600 * 4 + 600 * 5 * 14 = 44999

This does not fit as is into a signed 2 byte integer (-32768..32767). It fits into 2 byte unsigned integer (0..65535).

If you have to fit it into a signed integer, then easiest is to do calculcation in 32 bits, then substract (44999-32768) from the result, so it will fit into range of 2-byte signed integer. Then before decoding do reverse.

And for the opposite, you need division and reminder:

first = result % range_of_first + 1 // +1 to get values 1..600
second = result / range_of_first % range_of_second
third = result / range_of_first / range_of_second // % range_of_third redundant

Above all numbers are assumed to be non-negative, and usual integer division rounding is assumed, that is just chopping off any decimals.