Calculate max Data

Question

Over the last few days, I have been dealing in detail with numbers and number ranges, as well as with the conversion of a value into different number ranges.

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[X] Some information:

• I got a dataField that includes an specific amount of dataBlobs (int N like e.g 10 or 102, ...).

• Every single dataBlob can hold 16 different combination options.

• I got a dataString like "102" or "10101010" I want to encode in the data field.

• The amount of unique characters in dataString will be the baseValue I'm encoding from.
  
  ( If dataString is like "101010101" [2 unique chars "1", "0"] I will encode dataString from base 2 into base 16)
  
  ( If dataString is like "102" [3 unique chars "1", "0", "2"] I will encode dataString from base 3 into base 16)

• The encoding of the dataString into Base16 will give back a array like ["0", "15", "4", "7", "1", "1", ...] (all values between 0 and 15) and for every element in the encodedData I will add one Blob with the specific dataValue (see in the image below)

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[X] My question:

I'd like to calculate the maximum length of the dataString a dataField with N dataBlobs(16combinations) can hold.

What is given:

• The maximum amount of dataBlobs can used (N)
• The count of unique chars in dataString ( = the base-value I will encode from)

dataString = "10101010" | (from BASE 2 to BASE 16) will get [10, 10] --> Needs 2x dataBlobs

dataString = "102" | (from BASE 3 to BASE 16) will get [6] --> Needs 1x dataBlob

dataString = "2143567908" | (from BASE 10 to BASE 16) will get [7, 15, 12, 4, 4, 0, 2, 4] --> Needs 8x dataBlobs

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How can I calculate the amount of characters dataString (with a given Base value) can hold if I want to use N dataBlobs?

Answer 1

In general, the number of digits used to encode a nonnegative integer number $k$ in base $b$ is equal to $ \left \lceil\log_b k \right \rceil + 1$. For instance, to encode 32 in base 2, you need $\lceil \log_2 32\rceil + 1 = 5 + 1 = 6$ digits: $32_2 = 100000$.

If you want to use $N$ dataBlobs, you can encode a number that is at most equal to $16^N - 1$. It follows then that using base $b$ you will need a dataString at most $\left\lceil \log_b \left(16^N -1\right)\right\rceil + 1$ characters long.