Correction: Camazón Portela, D.; López Ramos, J.A. Error-Correcting Codes on Projective Bundles over Deligne–Lusztig Varieties. Mathematics 2023, 11, 3079

In the published publication [1], there was update regarding Daniel Camazón Portela’ affiliation, the correct one should be: Department of Mathematics, University of Almería, 04120 Almería, Spain; danielcp@ual.es.

The authors wish to make corrections on the result exposed in [1] concerning the dimension of the obtained codes. Let us recall the Corollaries:

Corollary 2.

Let $S_1$ be the Deligne–Lusztig surface of type $A_2$ defined over the field ${\mathbb{F}}_q$. Consider some b, such that $0<b<(q+1)$, and $V_i={\mathcal{O}}_{S_1}(n_{i}H-{\sum }_{j=1}^{q^2+q+1}{m}_{i,j}{B}_j)$ for $i=1,2$, where $H={\pi }^{*}({\mathcal{O}}_{{\mathbb{P}}^2}(1))$, verifying, for any pair, $i_1,i_2\in \mathbb{N}$ with $i_1+i_2=b$:

1. 

$(i_1{n}_1+i_2{n}_2)\le 3(q-1)$;

2. 

$(i_1{n}_1+i_2{n}_2)\ge {\sum }_{j=1}^3(i_1{m}_{1,j}+i_2{m}_{2,j})$;

3. 

$(i_1{m}_{1,j}+i_2{m}_{2,j})\ge (i_1{m}_{1,j+1}+i_2{m}_{2,j+1})$;

4. 

$3(i_1{n}_1+i_2{n}_2)>{\sum }_{j=1}^{q^2+q+1}(i_1{m}_{1,j+1}+i_2{m}_{2,j+1})$.

Let $T_1$ be the projective bundle $P(V_1\oplus V_2)$ over $S_1$, $p_1:T_1\to S_1$. Then, we can construct a code on $T_1$ over ${\mathbb{F}}_q$ with parameters

1. 

$n=(q^2+q+1){(q+1)}^2$;

2. 

$k={\sum }_{\begin{array}{c}{i}_1,i_2\in {\mathbb{N}}^2\\ i_1+i_2=b\end{array}}{\displaystyle \frac{1}{2}}((i_1{n}_1+i_2{n}_2)((i_1{n}_1+i_2{n}_2)+3)-{\sum }_{j=1}^{q^2+q+1}(i_1{m}_{1,j}+i_2{m}_{2,j})((i_1{m}_{1,j}+i_2{m}_{2,j})+1))+1$;

3. 

$d\ge n-(q^2+q+1)(q+1)b$.

Corollary 3.

Let $S_2$ be the Deligne–Lusztig surface of type ${}^{2}A_4$ defined over the field ${\mathbb{F}}_{q^2}$. Consider $V_i={\pi }^{*}{i}^{*}{\mathcal{O}}_{{\mathbb{P}}^4}(t_i)$ for $i=1,2$, and let $T_2$ be the projective bundle $P(V_1\oplus V_2)$ over $S_2$, $p_2:T_2\to S_2$. Then, for some $0<b<(q^2+1)$ and for any $t_1,t_2\in \mathbb{N}$ such that $(q+1)<(i_1{t}_1+i_2{t}_2)<(q^3+1)$ for any pair $i_1,i_2\in \mathbb{N}$ with $i_1+i_2=b$, we can construct a code on $T_2$ over ${\mathbb{F}}_{q^2}$ with parameters

1. 

$n=(q^5+1)(q^3+1){(q^2+1)}^2$;

2. 

$k={\sum }_{\begin{array}{c}{i}_1,i_2\in {\mathbb{N}}^2\\ i_1+i_2=b\end{array}}\left({\displaystyle \genfrac{}{}{0pt}{}{4+i_1{t}_1+i_2{t}_2}{i_1{t}_1+i_2{t}_2}}\right)-\left({\displaystyle \genfrac{}{}{0pt}{}{4+i_1{t}_1+i_2{t}_2-(q+1)}{i_1{t}_1+i_2{t}_2-(q+1)}}\right)$;

3. 

$d\ge n-(q^5+1)(q^3+1)(q^2+1)b$.

Instead of considering general vector bundles of rank 2 of the form $V=V_1\oplus V_2$, we should work with normalized vector bundles, that is, vector bundles $\tilde{V}$, verifying $H^0(S_i,\tilde{V})\ne 0$, but $H^0(S_i,\tilde{V}\otimes N)=0$ for $i=1,2$ and for any line bundle N with $deg(N)<0$. We insert Corollarys 4 and 5 before Example 2.

Corollary 4.

Let $S_1$ be the Deligne–Lusztig surface of type $A_2$ defined over the field ${\mathbb{F}}_q$. Let us consider $\tilde{V}={\mathcal{O}}_{S_1}\oplus {\mathcal{O}}_{S_1}(n_{1}H-{\sum }_{j=1}^{q^2+q+1}{m}_{1,j}{B}_j)$, with $m_{1,j}\ge 0$ for $j=1,...,q^2+q+1$ and $H^0(S_1,{\mathcal{O}}_{S_1}(n_{1}H-{\sum }_{j=1}^{q^2+q+1}{m}_{1,j}{B}_j)=0$, and let $T_1$ be the projective bundle $P(\tilde{V})$ over $S_1$, $p_1:T_1\to S_1$. Now, let L be the line bundle over $T_1$$L={\mathcal{O}}_{T_1}(b_1)\otimes p_1^{*}{\mathcal{O}}_{S_1}(D_1)$, where $D_1=n_{2}H-{\sum }_{j=1}^{q^2+q+1}{m}_{2,j}{B}_j$, with $n_2\le 3(q-1)$, $m_{2,j}\ge 0$ and $m_{1,j}{m}_{2,j}\ne 0$ for $j=1,...,q^2+q+1$. Then, we can construct a code on $T_1$ over ${\mathbb{F}}_q$ with parameters

1. 

$n=(q^2+q+1){(q+1)}^2$,

2. 

$k=h^0(S_1,{\mathcal{O}}_{S_1}(n_{2}H-{\sum }_{j=1}^{q^2+q+1}{m}_{2,j}{B}_j))+{\sum }_{t=1}^{b_1}{h}^0(S_1,{\mathcal{O}}_{S_1}((n_{1}t+n_2)H-$${\sum }_{j=1}^{q^2+q+1}{m}_{1,j}t+m_{2,j})B_j))$,

3. 

$d\ge (q^2+q+1){(q+1)}^2-(q^2+q+1)(q+1)b_1-{\sum }_{j=1}^{q^2+q+1}(b_1{m}_{1,j}+m_{2,j})(q+1)$.

Proof. 

Let $T_{1,B_j}=P({\mathcal{O}}_{B_j}\oplus {\mathcal{O}}_{B_j}(m_{1,j}))$. Then, there exists an isomorphism $\varphi :\widetilde{T_{1,B_j}}=P({\mathcal{O}}_{B_j}\oplus {\mathcal{O}}_{B_j}(-m_{1,j}))\to P({\mathcal{O}}_{B_j}\oplus {\mathcal{O}}_{B_j}(m_{1,j}))$. Now, let us consider the restriction ${L|}_{T_{1,B_j}}={\mathcal{O}}_{T_{1,B_j}}(b_1)\otimes p_1{|}_{T_{1,B_j}}^{*}{\mathcal{O}}_{B_j}(m_{2,j})$. We know that ${\varphi }^{*}(L{|}_{T_{1,B_j}})={\mathcal{O}}_{\widetilde{T_{1,B_j}}}(b_1)\otimes {\pi }^{*}{\mathcal{O}}_{B_j}(-b_1{m}_{1,j}+m_{2,j})$. Moreover, let us consider the divisor $H_1=\varsigma +m_{1,j}f\in A^1(\widetilde{T_{1,B_j}})$ that is nef by ([20], Chap. V.2). Then, by ([15], Corollary 3.2.), we know that

$$l_{T_{1,B_j}}\le {\displaystyle \frac{H_1{·L|}_{T_{1,B_j}}}{min\left\{H_1·f\right\}}}=b_1{m}_{1,j}+m_{2,j},$$

(50)

so, as a consequence of ([15], Proposition 3.2), it follows that

$$d\ge (q^2+q+1){(q+1)}^2-(q^2+q+1)(q+1)b_1-\sum _{j=1}^{q^2+q+1}(b_1{m}_{1,j}+m_{2,j})(q+1).$$

(51)

□

Corollary 5.

Let $S_2$ be the Deligne–Lusztig surface of type ${}^{2}A_4$ defined over the field ${\mathbb{F}}_{q^2}$. Let us consider $\tilde{V}={\mathcal{O}}_{S_2}\oplus {\mathcal{O}}_{S_2}(-n_1{H}_Z)$, where ${\mathcal{O}}_{S_2}(n{H}_Z)={\rho }^{*}{i}^{*}{\mathcal{O}}_{{\mathbb{P}}^4}(n)$, and let $T_2$ be the projective bundle $P(\tilde{V})$ over $S_2$, $p_2:T_2\to S_2$. Now, let L be the line bundle over $T_2$$L={\mathcal{O}}_{T_2}(b_2)\otimes p_2^{*}{\mathcal{O}}_{S_2}(D_2)$, with $D_2=n_2{H}_Z$. Then, we can construct a code on $T_2$ over ${\mathbb{F}}_{q^2}$ with parameters

1. 

$n=(q^5+1)(q^3+1){(q^2+1)}^2$,

2. 

$k=h^0(S_2,{\mathcal{O}}_{S_2}(n_2{H}_Z))+{\sum }_{t=1}^{b_2}{h}^0(S_2,{\mathcal{O}}_{S_2}((n_2-n_{1}t)H_Z))$,

3. 

$d\ge (q^5+1)(q^3+1){(q^2+1)}^2-(q^5+1)(q^3+1)(q^2+1)b_2-(q^5+1)(q^3+1)(2b_2{n}_1+n_2)(q^2+1)$.

Proof. 

Let $T_{2,B_j}=P({\mathcal{O}}_{B_j}\oplus {\mathcal{O}}_{B_j}(-n_1))$. Now, let us consider the restriction ${L|}_{T_{2,B_j}}={\mathcal{O}}_{T_{2,B_j}}(b_2)\otimes p_2{|}_{T_{2,B_j}}^{*}{\mathcal{O}}_{B_j}(n_2)$. Moreover, let us consider the divisor $H_2=\varsigma +n_{1}f\in A^1(T_{2,B_j})$ that is nef by ([20], Chap. V.2). Then by ([15], Corollary 3.2.) we know that

$$l_{T_{2,B_j}}\le {\displaystyle \frac{H_2{·L|}_{T_{2,B_j}}}{min\left\{H_2·f\right\}}}=2b_2{n}_1+n_2,$$

(52)

so, as a consequence of ([15], Proposition 3.2), it follows that

$$d\ge (q^5+1)(q^3+1){(q^2+1)}^2-(q^5+1)(q^3+1)(q^2+1)b_2-(q^5+1)(q^3+1)(2b_2{n}_1+n_2)(q^2+1).$$

(53)

□

In [1] (Examples 2 and 3), we compute the parameters of some of the codes in Corollary 2 and Corollary 3 for the binary case, $q=2$. However, the dimensions are not the correct ones, so we instead propose the following new examples of some of the codes in Corollary 4 and Corollary 5. We insert Example 4 after Example 3.

Example 2.

Let us consider the particular case $q=2$, with $b_1=1$, $n_1=1$, $n_2=3$, $m_{1,j}=1$ for $j=1,...,7$, $m_{2,j}=1$ for $j=1,2,3$ and $m_{2,j}=0$ otherwise, for the family of codes presented in Corollary 4. Then, we obtain codes with the following parameters:

1. 

$n=63$,

2. 

$k=7$,

3. 

$d\ge 12$.

Example 3.

For $q=2$, let us consider the particular case $b_2=1$, $n_1=1$, $n_2=1$ for the family of codes presented in Corollary 5. Then, we obtain a code with the following parameters:

1. 

$n=7425$,

2. 

$k=6$,

3. 

$d\ge 1485$.

Example 4.

Let us consider the particular case $q=3$, with $b_1=1$, $n_1=1$, $n_2=7$, $m_{1,j}=1$ for $j=1,...,13$, $m_{2,j}=1$ for $j=1,2,3$ and $m_{2,j}=0$ otherwise, for the family of codes presented in Corollary 4. Then, we obtain codes with the following parameters:

1. 

$n=208$,

2. 

$k=33$,

3. 

$d\ge 92$.

Note that whereas the performance of the family of codes presented in Corollary 4 is good in terms of the information rate $k/n$ and the minimum distance d, this is not the case for the family constructed in Corollary 5, where some modifications should be introduced in order to improve its information rate.

The authors state that the scientific conclusions are unaffected. This correction was approved by the Academic Editor. The original publication has also been updated.