Baseparitys

Baseparitys is a theoretical construct in number representation and coding theory that refers to the parity properties of digit sums across numeral bases. The concept collects a family of parity indicators obtained by expressing integers in different bases, forming a multi-base view of parity rather than a single-base measure.

For a base b > 1 and a nonnegative integer n, the baseparity p_b(n) is defined as the parity (0 for even, 1 for odd) of the sum of the digits of n in base b. The vector of baseparities across chosen bases forms the baseparitys of n. For example, n = 23 has base-10 representation 23, digit sum 2+3=5, so p_10(23) = 1; in base 2, 23 is 10111 with digit sum 4, so p_2(23) = 0; in base 3, 23 is 212 with digit sum 5, so p_3(23) = 1.

Properties: Unlike parity under addition, p_b is not preserved under general addition; its behavior depends on carry operations in base b. Taking parity across multiple bases can provide a lightweight, multi-base check that is more sensitive to certain digit errors than a single-base parity. However, baseparity checks are not cryptographically secure and can be affected by particular error patterns.

Applications: used in teaching to illustrate base representations, in simple error-detection schemes for data encoding, and as a research construct to study how digit sums behave under base changes.

See also: Parity, Digit sum, Base representation, Error detection. Notes: The term baseparitys is used in some expository discussions and may appear sporadically in literature; it is not a standard term in formal coding theory.