Base adder, given base and two numbers of that base, it adds them
The limit of the maximum radix of 26 sounds odd, why not use the full alphabet of characters up to 10 digits, and 26 letters, giving a total of radix-36?

You have not included a reinventing-the-wheel tag, and thus I can suggest the following:

public static String add (String num1, String num2, int base) {
    if (base &gt; 26) {
       // arbitrary limitation to 26... BigInt supports to 36.
    }
    BigInteger bione = new BigInteger(num1, base);
    BigInteger bitwo = new BigInteger(num2, base);
    return bione.add(bitwo).toString(base);
}


I believe that satisfies the review, optimization and best-practices request.