Binary duadic code[1]

Description

Member of a pair of cyclic linear binary codes that satisfy certain relations, depending on whether the pair is even-like or odd-like duadic. Duadic codes exist for lengths \(n\) that are products of powers of primes, with each prime being \(\pm 1\) modulo \(8\) [2].

Duadic codes come in two pairs, an even-like duadic pair and an odd-like duadic pair. All codewords in the respective pairs are even-like, i.e., \(\sum_i c_i = 0\), or odd-like, i.e., \(\sum_i c_i = 1\). A code with all even-like (odd-like) codewords is called even-like (odd-like).

Duadic code pairs can be defined in terms of their idempotent generators (see Cyclic-to-polynomial correspondence). A pair of even-like codes \(C_1\) and \(C_2\) with respective idempotents \(e_1\) and \(e_2\) is an even-like duadic pair if (1) \(e_1(x)+e_2(x)=1-\frac{1}{n}(1+x+x^2+\cdots+x^{n-1})\) and (2) there exists a multiplier \(\mu\) such that \(C_1 \mu=C_2\) and \(C_2 \mu=C_1\).

There is an odd-like duadic pair \(\{D_1,D_2\}\) associated with the even-like pair \(\{C_1, C_2\}\), where \(1-e_2(x)\) generates \(D_1\) and \(1-e_1(x)\) generates \(D_2\). The even-pair codes are \([n,\frac{n-1}{2}]\) codes while the odd-pair codes are \([n,\frac{n+1}{2}]\) codes.

Families of binary duadic codes were constructed in Ref. [3].