Theoretical Computer Science
Constructions for Permutation Codes in Powerline Communications
Designs, Codes and Cryptography
Irreducible polynomials over finite fields with prescribed trace/prescribed constant term
Finite Fields and Their Applications
On the number of primitive polynomials over finite fields
Finite Fields and Their Applications
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The trace of a degree n polynomial f(x) over GF(q) is the coefficient of xn-1. Carlitz [Proc. Amer. Math. Soc., 3 (1952), pp. 693--700] obtained an expression Iq(n,t) for the number of monic irreducible polynomials over GF(q) of degree n and trace t. Using a different approach, we derive a simple explicit expression for Iq(n,t). If t0, Iq(n,t) = (\sum \mu(d) q^{n/d})/(qn)$, where the sum is over all divisors d of n which are relatively prime to q. This same approach is used to count Lq(n,t), the number of q-ary Lyndon words whose characters sum to t mod q. This number is given by Lq(n,t) = (\sum {\rm gcd}(d,q) \mu(d) q^{n/d})/(qn)$, where the sum is over all divisors d of n for which gcd(d,q)|t. Both results rely on a new form of Möbius inversion.