Determining when the absolute state complexity of a Hermitian code achieves its DLP bound (Q2784499)

This article provides a detailed analysis of the absolute state complexity of an algebraic geometric code \(C=C_L(D,mQ_{\infty})\) constructed on an Hermitian curve \(H\) defined over \(\mathbb{F}_{q^2}\), where \(D\) is the sum of all \(q^3\) affine points on \(H\) and \(Q_{\infty}\) is its point at infinity. The absolute state complexity \(s[C]\) of a code \(C\) gives a measure of its decoding complexity.NEWLINENEWLINENEWLINEThe main tool used in the article is the DLP lower bound on \(s[C]\), introduced by \textit{G. D. Forney} [IEEE Trans. Inf. Theory 40, 1741-1752 (1994; Zbl 0826.94019)]. This bound was computed for Hermitian codes in an earlier paper of the same authors [Des. Codes. Cryptography 25, No. 1, 95-115 (2002; Zbl 1005.94026)]. Here they first determine when this bound is tight. In the other cases they give an improvement on it. In this way the authors can compute \(s[C]\) for most (but not all) Hermitian codes.