Earlier quoted context omitted.
My personal favorite encoding is base58 aka Bitcoin address encoding. It uses characters [A-Za-z0-9] except for [0OIl]. It is almost as dense as base64, "double clickable", but not (as) predictable in length as base32. It was chosen to avoid a number of the most annoying ambiguous letter shapes for hand-entry of long address strings. https://en.bitcoin.it/wiki/Base58Check_encoding
Is there a good reason why Base63 (nopad) doesn't exist? Ie Base64 minus the `-`, so that you almost get the density of base64 (nopad) but the double click friendly feature. I was reviewing encodings recently and didn't want to drop all the way down to base32, but for some reason the library i was using didn't allow anything beyond base32 and bas64 variants, despite having a feature where you can define your own base…
Yes, the reason is that you need 64 characters if you want each character to encode 6 bits as log2(64) == 6. If you only have 63 characters in your alphabet then one of your 6-bit combinations has no character to represent it.
Base32 can represent 5 bits per character because log2(32) == 5. Anything in between 32 and 64 doesn't buy you anything because there is no integer between 5 and 6.