Mass formula for self-dual codes over $$\varvec{\mathbb {F}}_q+u\varvec{\mathbb {F}}_q+u^2\varvec{\mathbb {F}}_q$$ F q + u

Self-dual codes over $\mathbb{F}_{q}+u\mathbb{F}_{q}+u^2\mathbb{F}_{q}$ and applications

“…Let C be a linear code of length n over F q + uF q + u 2 F q of type {k, l, m} and let h = n − (k + l + m). Using argument similar to those in Section 2 of [3], it can be deduced that the Hermitian dual C ⊥H of C is of type {h, m, l} and it contains q 3h+2m+l codewords. It follows that k = h and l = m if C is Hermitian self-dual.…”

“…For each positive integer n, let N H e (q, n) denote the number of distinct Hermitian self-dual linear codes of length n over F q + uF q + • • • + u e−1 F q . By extending techniques introduced in [3], the characterization and the number N H 3 (q, n) of Hermitian self-dual linear codes of length n over F q + uF q + u 2 F q are established.…”

“…By extending the technique used in the study of Euclidean self-dual linear codes over F q + uF q + u 2 F q in [3], complete characterization and enumeration of Hermitian self-dual linear codes over F q +uF q +u 2 F q have been established for all square prime powers q. An algorithm for constructions of such self-dual codes has veen provided as well as an illustrative example.…”

“…In the special case where H ∼ = A × Z 3 with 3 |A|, characterization and enumeration of H-quasi-abelian codes and self-dual H-quasi-abelian codes in F 3 m [H × B] have been completely determined for all finite abelian group B. As applications, characterization and enumeration of self-dual A × Z 3 -quasi-abelian codes in F 3 m [A × Z 3 × B] can be presented in terms of linear codes and self-dual linear codes over some extensions of F 3 m + uF 3 m + u 2 F 3 m determined in [3], [4] and Section 3.…”

“…In [16], the mass formula for Euclidean self-dual linear codes over Z p 3 has been studied. Characterization and enumeration of Euclidean self-dual linear codes over the ring F q + uF q + u 2 F q with u 3 = 0 have been given in [3].…”