Correction to: Efficient Search for Superspecial Hyperelliptic Curves of Genus Four with Automorphism Group Containing ${\textbf{C}}_6$

Correction to: Math.Comput.Sci. 2023 17:21 https://doi.org/10.1007/s11786-023-00571-w

Unfortunately, the below corrections are missed in the original Paper and the corrections are now updated.

In Sec 1, 3${\textrm{rd}}$ Paragraph, 7${\textrm{th}}$ Line, with complexity $\tilde{{O}}(p^{3})$ should be with complexity $O(p^{3})$

In Sec 2, 2nd Paragraph, 2${\textrm{nd}}$ Line, “and $Q_{8}$ the” should be “and ${\textbf{Q}}_{8}$ the”

In 2nd Paragraph under Remark 2.1.3, 9${\textrm{th}}$ Line, [24, Table 3] should be [24, Table 6]

In Theorem 2.1.5, 1st Line, $\overline{\textrm{Aut}(H)}$ should be $\overline{\textrm{Aut}}(H)$.

In Theorem 2.1.5, 3rd Line, $\overline{\textrm{Aut}(H)}$ should be $\overline{\textrm{Aut}}(H)$.

In Sec 2.2, 2nd Paragraph, 1st Line, “general curve” should be “general curve C of genus g over an algebraically closed field k”.

In Sec 2.2, 2nd Paragraph, 4th Line, “over K” should be “over k”.

In Sec 2.2, 2nd Paragraph, 5th Line, “K(x)” should be “k(x)”.

In Sec 2.2, 2nd Paragraph under Example 2.2.3, Equation (2.2.5), $U_{k}$ should be $U_{\ell }$

In Sec 2.2, 2nd Paragraph under Example 2.2.3, Equation $A(\ell )$, add 1 just above $r_{1}(\ell )$.

In Sec 2.2, 2nd Paragraph under Example 2.2.3, 7th Line under Equation A($\ell$), “the value of k” should be “the value of $\ell$”.

In Sec 2.4, 1st Paragraph under Theorem 2.4.4, 2nd Line, “Table 2 below” should be “Table 2”.

In In Sec 3, 1st Paragraph, 7th Line, “Let iota” should be “Let $\iota$” and “of of” should be “of”.

In Sec 3, 1st Paragraph, 8th Line, “has order-3” should be “has an order-3”.

In In Sec 3, 1st Paragraph, 8th Line, $-\zeta _{3}^{2}y$ should be $\zeta _{3}^{2}y$.

In Sec 3, 1st Paragraph, 9th Line, $\lambda = -\zeta _{3}^{2}$ should be $\lambda = \zeta _{3}^{2}$.

In Sec 3, 1st Paragraph, 9th Line, sigma$_{3}$ should be $\sigma _{3}$ and “is an has order-6, automorphism say” should be “has order-6, say”.

In Sec 3.1, 1st Paragraph, 7th Line, “a and b” should be “A and B”.

In the proof of Lemma 3.1.1, 1st Paragraph 2nd Line, $\zeta :\zeta _{3}$ should be $\zeta :=\zeta _{3}$

In the proof of Lemma 3.1.1, 2nd Paragraph, 1${\textrm{st}}$ Line, since ${\mathbb {C}}_{9}$ should be since ${{\textbf {C}}}_{9}$

In the proof of Lemma 3.1.2, 5th Line, remove an extra indent before “where $\mu$’ is an element ...”.

In Remark 3.2.2, 4th Paragraph, 2${\textrm{nd}}$ Line, $\sigma ^{\circ }\iota$ should be $\sigma _{3}^{\circ }\iota$

In Remark 3.2.2, 2nd Paragraph, 5th Line, $H_{a,b}/\langle \sigma \circ \iota \rangle (x : y : z)$ should be $H_{a,b}/\langle \sigma \circ \iota \rangle ;(x : y : z)$.

In Remark 3.2.2, 3rd Paragraph, 3rd Line, “$d_{4} = 0$ and” should be “$d_{4} = 0,$ and” (Oxford comma).

In Remark 3.2.2, 3rd Paragraph, 4th Line, “$\{ w^{(3)} \}$ and” should be “$\{ w^{(3)} \}$, and” (Oxford comma), and“$D_3$ and” should be “$D_3$, and” (Oxford comma).

In Remark 3.2.2, 4th Paragraph, 2nd Line, “(4, 3) and” should be “(4, 3), and” (Oxford comma).

In Remark 3.2.2, 5th Paragraph, 2nd Line, “(3, 1) and” should be “(3, 1), and” (Oxford comma), and “(1, 4)-th ones” should be “(4, 1)-th ones”.

In Theorem 4.1.1, 2nd Line, $mathbbF_{p^2}$ should be $\mathbb {F}_{p^{2}}$

In the proof of Theorem 4.1.1, 1st Paragraph, 10th Line, degree $\le$ D should be degree $\le D$

In the proof of Theorem 4.1.1, 1st Paragraph, 11th Line, tildeO, $(D^{2})$ should be $\tilde{{O}}(D^{2})$

In In the proof of Lemma 4.1.2, 1st Paragraph, 3rd Line, $\frac{(2i-1)p+(2j-1)}{2}$ should be $\frac{(2i-1)p-(2j-1)}{2}$

In the proof of Lemma 4.1.2, 1st Paragraph, 5th Line, $(g^{n})_{k-3}$ should be $(g^n)_{\ell - 3}$

In the proof of Lemma 4.1.2, 1st Paragraph, 7th Line, $\frac{(2i-1)p+(2j-1)}{2}$ should be $\frac{(2i-1)p-(2j-1)}{2}$

In the proof of Lemma 4.1.2, 2nd Paragraph, 1st Line, $\frac{(2i-1)p+(2j-1)}{2}$ should be $\frac{(2i-1)p-(2j-1)}{2}$

In the proof of Lemma 4.1.2, 2nd Paragraph, 4th Line, $ell\in 3{\mathbb {Z}}$ should be $\ell \in 3 \mathbb {Z}$

In the proof of Lemma 4.1.2, 2nd Paragraph, 5th Line, “$9n-\ell = 3p$ ,” (extra space before period) should be “$9n - \ell = 3p,$”, “$\ell = 3n - 3$ .” (extra space before period) should be “$\ell = 3n - 3.$”, $(g_n)\ell$ should be $(g^n)_{\ell }$, and $\ell \in 3\mathbb {Z}$ should be $\ell \in 3\mathbb {Z}$.

In the proof of Lemma 4.1.2, 2nd Paragraph, 6th Line, $3p = 6n+3< \ell < 9n$ should be $3p = 6n + 3 \le \ell \le 9n$.

In Remark 4.1.3, 8th Line, “Compute the roots of” should be “Compute the roots in $\mathbb {F}_{p^2}$ of”.

In Remark 4.1.3, 13th Line, “Compute the roots of” should be “Compute the roots in $\mathbb {F}_{p^2}$ of”.

Pages 17–18: The references in alphabetical order with respect to the last names of the first authors. Some extra commas should be removed.

In Reference “Algorithmic study of superspecial hyperelliptic curves over finite fields”, volumes, pages, and year (Vol. 70, 49–64, 2022) should be added.

Throughout the paper, l changes to $\ell$.