Correction to: Cyclic codes over $M_{4} (\mathbb {F}_{2}+u\mathbb {F}_{2})$  

Correction to: Cryptography and Communications

https://doi.org/10.1007/s12095-022-00572-9

• 1. Page 5, line 157, 158, 159: Equation is not printed properly, i.e., the representation for ξ should be in continuation. Moreover, font size is not matched with previous as well as later. The correct format is as below:

  $$\begin{array}{@{}rcl@{}} \xi&=& 8\deg(G_{1})+6\deg(G_{2})+4\deg(G_{3})+2\deg(G_{4})+3\deg(G_{5})+2\deg(G_{6})+\deg(G_{7})\\ &&+4\deg(G_{8})+3\deg(G_{9})+3\deg(G_{10})+5\deg(G_{11})+2\deg(G_{12}). \end{array}$$

• 2. Page 6, line 183, 184, 185: Again same problem is there, i.e., the complete expression for ξ should be in continuation. Moreover, font size is not matched with previous as well as later. The correct format is given below:

  $$\begin{array}{@{}rcl@{}} \text{where } \xi&=& 8\deg(G_{1})+6\deg(G_{2})+4\deg(G_{3})+2\deg(G_{4})+3\deg(G_{5})+2\deg(G_{6})\\ &&+\deg(G_{7})+4\deg(G_{8})+3\deg(G_{9})+3\deg(G_{10})+5\deg(G_{11})+2\deg(G_{12}). \end{array}$$

• 3. Page 8, line 236-237: Equation is not printed properly. Moreover, font size is not matched with previous as well as later. Kindly put the correct format as given below:

  $$\begin{array}{@{}rcl@{}} C&=&\langle\hat{G_{1}}\rangle\oplus \langle v\hat{G_{2}}\rangle\oplus \langle v^{2}\hat{G_{3}}\rangle\oplus \langle v^{3}\hat{G_{4}}\rangle\oplus \langle uv\hat{G_{5}}\rangle\oplus \langle uv^{2}\hat{G_{6}}\rangle\oplus \langle uv^{3}\hat{G_{7}}\rangle\oplus \langle u\hat{G_{8}}\rangle\oplus (\langle uv^{2}\hat{G_{9}}\rangle \\ &&+ \langle v^{3}\hat{G_{9}}\rangle)\oplus \langle(uv+v^{2}h_{\alpha}+v^{3}h_{\beta}+uv^{2}h_{\delta})\hat{G_{10}}\rangle\oplus \langle (u+vh_{\alpha}+v^{2}h_{\beta}+v^{3}h_{\gamma} +uvh_{\delta}\\ &&+uv^{2}h_{\delta^{\prime}})\hat{G_{11}}\rangle\oplus \langle (uv^{2}+v^{3}h_{\alpha})\hat{G_{12}}\rangle, \end{array}$$

• 4. Page 8, line 241, 242, 243: Equation is not printed properly, i.e., whole η equation should be in continuation. The correct format is given below:

  $$\begin{array}{@{}rcl@{}} \eta&=& 8\deg(G_{0})+2\deg(G_{2})+4\deg(G_{3})+6\deg(G_{4})+5\deg(G_{5})+6\deg(G_{6})+7\deg(G_{7})\\ &&+4\deg(G_{8})+5\deg(G_{9})+5\deg(G_{10})+3\deg(G_{11})+6\deg(G_{12}). \end{array}$$

• 5. Page 8, line 245-246: Equation is not printed properly, i.e., whole ξ equation should be in continuation. Kindly put the correct format as given below:

  $$\begin{array}{@{}rcl@{}} \xi&=& 8\deg(G_{1})+6\deg(G_{2})+4\deg(G_{3})+2\deg(G_{4})+3\deg(G_{5})+2\deg(G_{6})+\deg(G_{7})\\ &&+4\deg(G_{8})+3\deg(G_{9})+3\deg(G_{10})+5\deg(G_{11})+2\deg(G_{12}). \end{array}$$

• 6. Page 9, line 247-248: Equation is not printed properly, i.e., whole n equation should be in continuation. Moreover, font size is not matched with previous as well as later. The correct format is given below:

  $$\begin{array}{@{}rcl@{}} n&=&\deg(G_{0})+ \deg(G_{1})+\deg(G_{2})+\deg(G_{3})+\deg(G_{4})+\deg(G_{5})+\deg(G_{6})+\deg(G_{7})\\ &&+\deg(G_{8})+\deg(G_{9})+\deg(G_{10})+\deg(G_{11})+\deg(G_{12}), \end{array}$$

• 7. Page 9, line 249-250: Equation is not printed properly, i.e., whole η = 8n − ξ equation should be in continuation. Moreover, font size is not matched with previous as well as later. The corrected format is given below:

  $$\begin{array}{@{}rcl@{}} \eta&=&8n-\xi= 8\deg(G_{0})+2\deg(G_{2})+4\deg(G_{3})+6\deg(G_{4})+5\deg(G_{5})+6\deg(G_{6})\\ &&+7\deg(G_{7})+4\deg(G_{8})+5\deg(G_{9})+5\deg(G_{10})+3\deg(G_{11})+6\deg(G_{12}). \end{array}$$

• 8. Page 9, line 251, 252, 253, 254: Equation is not printed properly. Moreover, font size is not matched with previous as well as later. The correct format is given below:

  $$\begin{array}{@{}rcl@{}} \text{Now, consider } C^{*}&=&\langle\hat{G_{0}^{*}}\rangle\oplus \langle v^{3}\hat{G_{2}^{*}}\rangle\oplus \langle v^{2}\hat{G_{3}^{*}}\rangle\oplus \langle v\hat{G_{4}^{*}}\rangle\oplus \langle(u+vh_{\alpha}+v^{2}h_{\beta}+v^{3}h_{\gamma} \\&&+uvh_{\delta}+uv^{2}h_{\delta^{\prime}}) \hat{G_{5}^{*}}\rangle\oplus (\langle u\hat{G_{6}^{*}}\rangle + \langle v^{2}\hat{G_{6}^{*}}\rangle)\oplus (\langle u\hat{G_{7}^{*}}\rangle + \langle v\hat{G_{7}^{*}}\rangle)\oplus\\ &&\langle u\hat{G_{8}^{*}}\rangle\oplus (\langle(uv+v^{2}h_{\alpha}+v^{3}h_{\beta}+uv^{2}h_{\delta})\hat{G_{9}^{*}}\rangle + \langle v^{3}\hat{G_{9}^{*}}\rangle)\oplus (\langle(uv\\ &&+v^{2}h_{\alpha}+v^{3}h_{\beta}+uv^{2}h_{\delta})\hat{G_{10}^{*}}\rangle+\langle uv^{2}\hat{G_{10}^{*}}\rangle + \langle v^{3}\hat{G_{10}^{*}}\rangle)\oplus (\langle uv^{3}\hat{G_{11}^{*}}\rangle\\ &&+\langle v^{2}\hat{G_{11}^{*}}\rangle )\oplus (\langle(u+vh_{\alpha}+v^{2}h_{\beta}+v^{3}h_{\gamma} +uvh_{\delta}+uv^{2}h_{\delta^{\prime}}) \hat{G_{12}^{*}}\rangle\\ &&+ \langle v^{3}\hat{G_{12}^{*}}\rangle). \end{array}$$

• 9. Page 9, line 267, 268, 269: Equation is not printed properly, i.e., whole η equation should be in continuation. Moreover, font size is not matched with previous as well as later. Kindly put the correct format as given below:

  $$\begin{array}{@{}rcl@{}} \eta&=& 8\deg(G_{0})+2\deg(G_{2})+4\deg(G_{3})+6\deg(G_{4})+5\deg(G_{5})+6\deg(G_{6})\\ &&+7\deg(G_{7})+4\deg(G_{8})+5\deg(G_{9})+5\deg(G_{10})+3\deg(G_{11})+6\deg(G_{12}). \end{array}$$

• 10. Page 10, line 291-292: Equation is not printed properly. Moreover, font size is not matched with previous as well as later. The correct format is given below:

  $$\begin{array}{@{}rcl@{}} C&=&\langle\hat{G_{1}}\rangle\oplus \langle v\hat{G_{2}}\rangle\oplus \langle v^{2}\hat{G_{3}}\rangle\oplus \langle v^{3}\hat{G_{4}}\rangle\oplus \langle uv\hat{G_{5}}\rangle\oplus \langle uv^{2}\hat{G_{6}}\rangle\oplus \langle uv^{3}\hat{G_{7}}\rangle\oplus \langle u\hat{G_{8}}\rangle\oplus (\langle uv^{2}\hat{G_{9}}\rangle\\ &&+ \langle v^{3}\hat{G_{9}}\rangle)\oplus \langle(uv+v^{2}h_{\alpha}+v^{3}h_{\beta}+uv^{2}h_{\delta})\hat{G_{10}}\rangle\oplus \langle (u+vh_{\alpha}+v^{2}h_{\beta}+v^{3}h_{\gamma}\\ &&+uvh_{\delta}+uv^{2}h_{\delta^{\prime}})\hat{G_{11}}\rangle\oplus \langle (uv^{2}+v^{3}h_{\alpha})\hat{G_{12}}\rangle, \end{array}$$

• 11. Page 10, line 296, 297, 298: Equation is not printed properly, i.e., whole η equation should be in continuation. Moreover, font size is not matched with previous as well as later. Kindly put the correct format as given below:

  $$\begin{array}{@{}rcl@{}} \eta&=& 8\deg(G_{0})+2\deg(G_{2})+4\deg(G_{3})+6\deg(G_{4})+5\deg(G_{5})+6\deg(G_{6})+7\deg(G_{7})\\ &&+4\deg(G_{8})+5\deg(G_{9})+5\deg(G_{10})+3\deg(G_{11})+6\deg(G_{12}). \end{array}$$

• 12. Page 12, line 350-351: Matrix is not printed properly and suggested corrections at online file are also not implemented. Kindly put the correct format as given below:

  $$\left( \begin{array}{cccc} a_{1}+ub_{1} & a_{2}+ub_{2}&a_{3}+ub_{3}&a_{4}+ub_{4}\\ (a_{2}+a_{5})+u(b_{2}+b_{5})& (a_{3}+a_{6})+u(b_{3}+b_{6})&(a_{4}+a_{7})+u(b_{4}+b_{7})&(a_{1}+a_{2}+a_{3}+a_{4}+a_{8})\\ &&&+u(b_{1}+b_{2}+b_{3}+b_{4}+b_{8})\\ a_{9}+ub_{9} & a_{10}+ub_{10}&a_{11}+ub_{11}&a_{12}+ub_{12}\\ (a_{10}+a_{13})+u(b_{10}+b_{13})& (a_{11}+a_{14})+u(b_{11}+b_{14})&(a_{12}+a_{15})+u(b_{12}+b_{15})&(a_{9}+a_{10}+a_{11}+a_{12}+a_{16})\\ &&&+u(b_{9}+b_{10}+b_{11}+b_{12}+b_{16}) \end{array}\right)$$
  $$\begin{array}{@{}rcl@{}} &&\rightarrow (a_{1}+a_{2}w+a_{3}w^{2}+a_{4}w^{3})+v(b_{1}+b_{2}w+b_{3}w^{2}+b_{4}w^{3})+v^{2}(a_{5}+a_{6}w+a_{7}w^{2}+a_{8}w^{3})\\ &&+v^{3}(b_{5}+b_{6}w+b_{7}w^{2}+b_{8}w^{3})+u(a_{9}+a_{10}w+a_{11}w^{2}+a_{12}w^{3})+uv(b_{9}+b_{10}w+b_{11}w^{2}+b_{12}w^{3})\\ &&+uv^{2}(a_{13}+a_{14}w+a_{15}w^{2}+a_{16}w^{3})+uv^{3}(b_{13}+b_{14}w+b_{15}w^{2}+b_{16}w^{3}) \end{array}$$

  where $a_i,b_j \in \mathbb{F}_{2} \;\mathrm{for} \; 1 \le i, j \le 16.$