COUNTING BINARY LCD CYCLIC CODES VIA NEGATION ORBITS: CLOSED FORMS AT LENGTHS $2^{m}-1$ AND $2^{m}+1$

Source abstract

We give an explicit negation orbit formulation of the standard Yang-Massey criterion for binary Euclidean LCD cyclic codes of odd length, counting both the zero code and the whole ambient space. With the defining set taken to be the full zero set of the generator polynomial, the LCD condition is $T=-T$, so the enumeration is equivalent to choosing unions of orbits of the negation involution $\imath(Q)=-Q$ on $\mathcal{C}_{2, n}$, the set of 2-cyclotomic cosets modulo $n$. The number of such codes is $$2^{\left|\mathcal{C}_{2, n} /\langle 1\rangle\right|},$$ and we record the corresponding divisor-sum expression in terms of $\operatorname{ord}_{d}(2)$ and an explicit self-pairing indicator for each divisor $d \mid n$. This is a binary cyclic specialization of known irreducible-factor enumerations and is used here only as setup. Our main result is a closed form enumeration at the Mersenne lengths $n=2^{m}-1$, as a function of $m$ alone. We prove $$N_{L C D}(m)=2^{1+\frac{1}{2}(T(m)+S(m))},$$ where $T(m)$ counts monic irreducible binary polynomials of degrees $e>1$ dividing $m$, and $S(m)$ counts the self-reciprocal ones; both are explicit Möbius sums over the divisors of $m$. In particular, the count requires no factorization of $2^{m}-1$, and it extends the odd-prime- $m$ formula of Li et al. [9] to all $m \geq 2$. The exponent satisfies $\log _{2} N_{L C D}(m)=2^{m-1} / m(1+o(1))$. For the lengths $n=2^{m}+1$, where every binary cyclic code is LCD, the number of (necessarily LCD) binary cyclic codes is $2^{1+F(m)}$ for an explicit Möbius sum $F(m)$. The same exponent appears in the $q=2$ case of the enumeration by Liu et al. [11] at the lengths $\lambda\left(q^{m}+1\right)$, under a convention that omits the ambient space code; we state this overlap explicitly. For $m \in\{3,4,5,6,7\}$, the Mersenne counts are 4, 16, 16, 256, 1024, and the closed form is verified against direct negation orbit enumeration for $m \leq 16$.