Correction to: Multi-twisted additive codes over finite fields

Correction to: Beitr Algebra Geom (2022) 63:287–320 https://doi.org/10.1007/s13366-021-00576-1

The purpose of this note is to rectify an error in equation (3.10), which provides the value of the Witt index $m_v$ of the non-degenerate quadratic space $({\mathcal {G}}_v, {\mathcal {Q}}_v)$ over ${\mathcal {F}}_{v}$ when q is odd, $v \in {\mathcal {J}}_{1},$ $\epsilon _vt$ is even and $\delta \in \{0,*\},$ (see the proof of Lemma 3.5(a) on page 311). In this case, the correct value of the Witt index $m_v$ is given by

$$\begin{aligned} m_v=\left\{ \begin{array}{ll} \epsilon _vt/2&{}\text {if }\epsilon _vt\equiv 2~(\text {mod }4)\text { and }q\equiv 3~(\text {mod }4);\\ (\epsilon _vt-2)/2&{} \text {if either }\epsilon _vt\text { is even and }q\equiv 1~(\text {mod }4)\\ &{} \text {or } \epsilon _vt\equiv 0~(\text {mod }4)\text { and }q\equiv 3~(\text {mod }4), \end{array}\right. \end{aligned}$$

(3.10)

(see page 279 of Huffman (2010) for more details). That is, in this case, the value of the Witt index $m_v$ specified in equation (3.10) of the published version should be flipped, as Theorem 1 of Pless (1968) is not applicable in this case. This error in the value of $m_v$ gives rise to flipping errors in the values of the numbers ${\mathfrak {M}}_{v}$ and ${\mathfrak {N}}_{v}$ obtained respectively in Lemma 3.5(a) (and hence in the statement of Theorem 3.2) and Lemma 3.8 (and hence in the statement of Theorem 3.3) when q is odd, $v \in {\mathcal {J}}_{1},$ $\epsilon _vt$ is even and $\delta \in \{0,*\}.$ The values of the numbers ${\mathfrak {M}}_{v}$ and ${\mathfrak {N}}_{v}$ are correct in the remaining cases. Besides this, this error in $m_v$ gives rise to minor changes in the statement and the proof of Lemma 3.7. The rest of the numbers and results obtained in this paper are correct. All proof techniques are also correct.

Below we list all the changes that will rectify these errors:

• 1. Page 309: In the statement of Theorem 3.2, when $v\in {\mathcal {J}}_1,$ q is odd, $\delta \in \{0,*\}$ and $\epsilon _vt$ is even, the value of ${\mathfrak {M}}_v$ should be replaced by

  $$\begin{aligned} {\mathfrak {M}}_v=\left\{ \begin{array}{ll} \sum \limits _{k=0}^{\epsilon _vt/2}\genfrac[]{0.0pt}{}{\epsilon _vt/2}{k}_{q}\prod \limits _{d=0}^{k-1}\left( q^{\frac{\epsilon _vt-2d-2}{2}}+1\right) &{} \text {if }\epsilon _vt\equiv 2~(\text {mod }4) \text { and } q\equiv 3~(\text {mod }4);\\ \sum \limits _{k=0}^{(\epsilon _vt-2)/2}\genfrac[]{0.0pt}{}{(\epsilon _vt-2)/2}{k}_{q}\prod \limits _{d=0}^{k-1}\left( q^{\frac{\epsilon _vt-2d}{2}}+1\right) &{} \text {if either }\epsilon _vt \text { is even and }q\equiv 1~(\text {mod }4)\\ {} &{}\text { or }\epsilon _vt\equiv 0~(\text {mod }4) \text { and }q\equiv 3~(\text {mod }4). \end{array}\right. \end{aligned}$$

  In the remaining cases, the values of ${\mathfrak {M}}_v$ will remain the same as in the original published version.

• 2. Page 310: In the statement of Lemma 3.5(a), when $v\in {\mathcal {J}}_1,$ q is odd, $\delta \in \{0,*\}$ and $\epsilon _vt$ is even, the value of ${\mathfrak {M}}_v$ should be replaced by

  $$\begin{aligned} {\mathfrak {M}}_v=\left\{ \begin{array}{ll} \sum \limits _{k=0}^{\epsilon _vt/2}\genfrac[]{0.0pt}{}{\epsilon _vt/2}{k}_{q}\prod \limits _{d=0}^{k-1}\left( q^{\frac{\epsilon _vt-2d-2}{2}}+1\right) &{} \text {if }\epsilon _vt\equiv 2~(\text {mod }4)\text { and }q\equiv 3~(\text {mod }4);\\ \sum \limits _{k=0}^{(\epsilon _vt-2)/2}\genfrac[]{0.0pt}{}{(\epsilon _vt-2)/2}{k}_{q}\prod \limits _{d=0}^{k-1}\left( q^{\frac{\epsilon _vt-2d}{2}}+1\right) &{} \text {if either }\epsilon _vt \text { is even and }q\equiv 1~(\text {mod }4)\\ &{} \text {or } \epsilon _vt\equiv 0~(\text {mod }4)\text { and }q\equiv 3~(\text {mod }4); \end{array} \right. \end{aligned}$$

  In the remaining cases, the values of ${\mathfrak {M}}_v$ will remain the same as in the original published version.

• 3. Page 310, the last line: In the proof of case I (i.e., when q is odd) of Lemma 3.5(a), “Further, by Theorem 1 of Pless (1968), we note that ...” should be replaced by “Further, by Huffman (2010, p. 279), we note that ...”.

• 4. Page 311: Equation (3.10) should be replaced by the following:

  $$\begin{aligned} m_v=\left\{ \begin{array}{ll} (\epsilon _vt-1)/2&{}\text {if }\epsilon _vt\text { is odd}; \\ \epsilon _vt/2&{}\text {if }\epsilon _vt\equiv 2~(\text {mod }4)\text { and }q\equiv 3~(\text {mod }4);\\ (\epsilon _vt-2)/2&{} \text {if either }\epsilon _vt \text { is even and }q\equiv 1~(\text {mod }4)\\ &{} \text {or } \epsilon _vt\equiv 0~(\text {mod }4)\text { and }q\equiv 3~(\text {mod }4). \end{array}\right. \end{aligned}$$

  (3.10)

  When $v \in {\mathcal {J}}_{1},$ $\delta \in \{0,*\},$ $\epsilon _v t$ is even, and q is odd, we see that $m_v=\epsilon _vt/2$ if and only if $\epsilon _vt\equiv 2~(\text {mod }4)$ and $q\equiv 3~(\text {mod }4),$ which holds if and only if $(-1)^{\epsilon _vt/2}$ is a non-square in ${\mathbb {F}}_q.$

• 5. Page 315, line 3: In the statement of Theorem 3.3, “...when $\delta \in \{0,*\},$ $(-1)^{\epsilon _vt/2}$ is a square in ${\mathbb {F}}_q$ for each $v \in {\mathcal {J}}_{1}$” should be replaced by “...when $\delta \in \{0,*\}$ and q is odd, the element $(-1)^{\epsilon _vt/2}$ is a non-square in ${\mathbb {F}}_q$ for each $v \in {\mathcal {J}}_{1}$”.

• 6. Page 315, lines 8-9: In the statement of Theorem 3.3, when $v\in {\mathcal {J}}_1,$ q is odd and $\delta \in \{0,*\},$ we assumed that $(-1)^{\epsilon _vt/2}$ is a non-square in ${\mathbb {F}}_q,$ which holds if and only if $\epsilon _vt\equiv 2~(\text {mod }4)$ and $q\equiv 3~(\text {mod }4).$ In this case, the value of ${\mathfrak {N}}_v$ should be replaced by the following:

  $$\begin{aligned} {\mathfrak {N}}_v=\prod \limits _{a=0}^{(\epsilon _vt/2)-1}\left( q^{\frac{\epsilon _vt-2a-2}{2}}+1\right) \text { when }\delta \in \{0,*\}, \epsilon _vt\equiv 2~(\text {mod }4)\text { and }q\equiv 3~(\text {mod }4). \end{aligned}$$

  In the remaining cases, the values of ${\mathfrak {N}}_v$ will remain the same as in the original published version.

• 7. Page 315: In the statement of Lemma 3.7, “(ii) the element $(-1)^{\epsilon _vt/2}$ is a square in ${\mathbb {F}}_q$ when $\delta \in \{0,*\}$ and $v \in {\mathcal {J}}_{1}$” should be replaced by “(ii) the element $(-1)^{\epsilon _vt/2}$ is a non-square in ${\mathbb {F}}_q$ when $\delta \in \{0,*\},$ q is odd and $v \in {\mathcal {J}}_{1}$”.

• 8. Page 315: The third paragraph in the proof of Lemma 3.7 should be replaced by the following: “When $v \in {\mathcal {J}}_{1},$ q is odd and $\delta \in \{0,*\},$ by Lemma 3.4(a), we note that $({\mathcal {G}}_v,[\cdot ,\cdot ]_\delta \upharpoonright _{{\mathcal {G}}_v\times {\mathcal {G}}_v})$ is an orthogonal space over ${\mathcal {F}}_v.$ Since $\epsilon _vt$ is even, we see, by (3.10), that the Witt index of $({\mathcal {G}}_v,[\cdot ,\cdot ]_\delta \upharpoonright _{{\mathcal {G}}_v\times {\mathcal {G}}_v})$ is $\epsilon _vt/2$ if and only if and $(-1)^{\epsilon _vt/2}$ is a non-square in ${\mathbb {F}}_q.$"

• 9. Page 315, line 33 (the last paragraph): “...in the case when $\delta \in \{0,*\},$ $(-1)^{\epsilon _vt/2}$ is a square in ${\mathbb {F}}_q$ for each $v \in {\mathcal {J}}_{1}$” should be replaced by “...in the case when $\delta \in \{0,*\}$ and q is odd, $(-1)^{\epsilon _vt/2}$ is a non-square in ${\mathbb {F}}_q$ for each $v \in {\mathcal {J}}_{1}$”.

• 10. Page 316: In the statement of Lemma 3.8, “...that $(-1)^{\epsilon _vt/2}$ is a square in ${\mathbb {F}}_q$ when $\delta \in \{0,*\}$ and $v \in {\mathcal {J}}_{1}$” should be replaced by “...that $(-1)^{\epsilon _vt/2}$ is a non-square in ${\mathbb {F}}_q$ when $\delta \in \{0,*\},$ q is odd and $v \in {\mathcal {J}}_{1}$”.

• 11. Page 316, line 5: In the statement of Lemma 3.8, when $v\in {\mathcal {J}}_1,$ q is odd and $\delta \in \{0,*\},$ we have assumed that $(-1)^{\epsilon _vt/2}\text { is a non-square in }{\mathbb {F}}_q,$ which holds if and only if $\epsilon _vt\equiv 2~(\text {mod }4)$ and $q\equiv 3~(\text {mod }4).$ In this case, the value of ${\mathfrak {N}}_v$ should be replaced by

  $$\begin{aligned} {\mathfrak {N}}_v=\prod \limits _{a=0}^{(\epsilon _vt/2)-1}\left( q^{\frac{\epsilon _vt-2a-2}{2}}+1\right) \text { if }\delta \in \{0,*\}, \epsilon _vt\equiv 2~(\text {mod }4)\text { and }q\equiv 3~(\text {mod }4). \end{aligned}$$

  In the remaining cases, the values of ${\mathfrak {N}}_v$ will remain the same as in the original published version.

• 12. Page 316, line 14: In the proof of Lemma 3.8(a), “From this point on, let $\delta \in \{0,*\}.$ Here $(-1)^{\epsilon _vt/2}$ is a square in ${\mathbb {F}}_q.$” should be replaced by “From this point on, let $\delta \in \{0,*\}.$ Here when q is odd, we know that $(-1)^{\epsilon _vt/2}$ is a non-square in ${\mathbb {F}}_q.$”

• 13. Page 316, line 21: In the proof of case I of Lemma 3.8(a), “Further, by Theorem 1 of Pless (1968), we note that ...” should be replaced by “Further, by (3.10), we note that ...".