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

Daniel Camazón Portela; Juan Antonio López Ramos

doi:10.3390/math12233634

and

Department of Mathematics, University of Almería, 04120 Almería, Spain

^*

Author to whom correspondence should be addressed.

^†

These authors contributed equally to this work.

Mathematics2024, 12(23), 3634;https://doi.org/10.3390/math12233634

This article belongs to the Special Issue New Advances in Algebra, Ring Theory and Homological Algebra, 2nd Edition

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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

F_{q}

. Consider some b, such that

0 < b < (q + 1)

, and

V_{i} = O_{S_{1}} (n_{i} H - \sum_{j = 1}^{q^{2} + q + 1} m_{i, j} B_{j})

for

i = 1, 2

, where

H = π^{*} (O_{P^{2}} (1))

, verifying, for any pair,

i_{1}, i_{2} \in N

with

i_{1} + i_{2} = b

:

1.: $(i_{1} n_{1} + i_{2} n_{2}) \leq 3 (q - 1)$ ;
2.: $(i_{1} n_{1} + i_{2} n_{2}) \geq \sum_{j = 1}^{3} (i_{1} m_{1, j} + i_{2} m_{2, j})$ ;
3.: $(i_{1} m_{1, j} + i_{2} m_{2, j}) \geq (i_{1} m_{1, j + 1} + i_{2} m_{2, j + 1})$ ;
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

F_{q}

with parameters

1.: $n = (q^{2} + q + 1) {(q + 1)}^{2}$ ;
2.: $k = \sum_{\begin{matrix} i_{1}, i_{2} \in N^{2} \\ i_{1} + i_{2} = b \end{matrix}} \frac{1}{2} ((i_{1} n_{1} + i_{2} n_{2}) ((i_{1} n_{1} + i_{2} n_{2}) + 3) - \sum_{j = 1}^{q^{2} + q + 1} (i_{1} m_{1, j} + i_{2} m_{2, j}) ((i_{1} m_{1, j} + i_{2} m_{2, j}) + 1)) + 1$ ;
3.: $d \geq n - (q^{2} + q + 1) (q + 1) b$ .

Corollary 3.

Let

S_{2}

be the Deligne–Lusztig surface of type

{}^{2}A_{4}

defined over the field

F_{q^{2}}

. Consider

V_{i} = π^{*} i^{*} O_{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 N

such that

(q + 1) < (i_{1} t_{1} + i_{2} t_{2}) < (q^{3} + 1)

for any pair

i_{1}, i_{2} \in N

with

i_{1} + i_{2} = b

, we can construct a code on

T_{2}

over

F_{q^{2}}

with parameters

1.: $n = (q^{5} + 1) (q^{3} + 1) {(q^{2} + 1)}^{2}$ ;
2.: $k = \sum_{\begin{matrix} i_{1}, i_{2} \in N^{2} \\ i_{1} + i_{2} = b \end{matrix}} (\binom{4 + i_{1} t_{1} + i_{2} t_{2}}{i_{1} t_{1} + i_{2} t_{2}}) - (\binom{4 + i_{1} t_{1} + i_{2} t_{2} - (q + 1)}{i_{1} t_{1} + i_{2} t_{2} - (q + 1)})$ ;
3.: $d \geq n - (q^{5} + 1) (q^{3} + 1) (q^{2} + 1) b$ .

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}) \neq 0

, but

H^{0} (S_{i}, \tilde{V} \otimes N) = 0

for

i = 1, 2

and for any line bundle N with

d e g (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

F_{q}

. Let us consider

\tilde{V} = O_{S_{1}} \oplus O_{S_{1}} (n_{1} H - \sum_{j = 1}^{q^{2} + q + 1} m_{1, j} B_{j})

, with

m_{1, j} \geq 0

for

j = 1, . . ., q^{2} + q + 1

and

H^{0} (S_{1}, 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 = O_{T_{1}} (b_{1}) \otimes p_{1}^{*} 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} \leq 3 (q - 1)

,

m_{2, j} \geq 0

and

m_{1, j} m_{2, j} \neq 0

for

j = 1, . . ., q^{2} + q + 1

. Then, we can construct a code on

T_{1}

over

F_{q}

with parameters

1.: $n = (q^{2} + q + 1) {(q + 1)}^{2}$ ,
2.: $k = h^{0} (S_{1}, O_{S_{1}} (n_{2} H - \sum_{j = 1}^{q^{2} + q + 1} m_{2, j} B_{j})) + \sum_{t = 1}^{b_{1}} h^{0} (S_{1}, O_{S_{1}} ((n_{1} t + n_{2}) H -$ $\sum_{j = 1}^{q^{2} + q + 1} m_{1, j} t + m_{2, j}) B_{j}))$ ,
3.: $d \geq (q^{2} + q + 1) {(q + 1)}^{2} - (q^{2} + q + 1) (q + 1) b_{1} - \sum_{j = 1}^{q^{2} + q + 1} (b_{1} m_{1, j} + m_{2, j}) (q + 1)$ .

Proof.

Let

T_{1, B_{j}} = P (O_{B_{j}} \oplus O_{B_{j}} (m_{1, j}))

. Then, there exists an isomorphism

ϕ : \tilde{T_{1, B_{j}}} = P (O_{B_{j}} \oplus O_{B_{j}} (- m_{1, j})) \to P (O_{B_{j}} \oplus O_{B_{j}} (m_{1, j}))

. Now, let us consider the restriction

{L |}_{T_{1, B_{j}}} = O_{T_{1, B_{j}}} (b_{1}) \otimes p_{1} |_{T_{1, B_{j}}}^{*} O_{B_{j}} (m_{2, j})

. We know that

ϕ^{*} (L |_{T_{1, B_{j}}}) = O_{\tilde{T_{1, B_{j}}}} (b_{1}) \otimes π^{*} O_{B_{j}} (- b_{1} m_{1, j} + m_{2, j})

. Moreover, let us consider the divisor

H_{1} = ς + m_{1, j} f \in A^{1} (\tilde{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}}} \leq \frac{H_{1} {\cdot L |}_{T_{1, B_{j}}}}{m i n \{H_{1} \cdot f\}} = b_{1} m_{1, j} + m_{2, j},

(50)

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

d \geq (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

F_{q^{2}}

. Let us consider

\tilde{V} = O_{S_{2}} \oplus O_{S_{2}} (- n_{1} H_{Z})

, where

O_{S_{2}} (n H_{Z}) = ρ^{*} i^{*} O_{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 = O_{T_{2}} (b_{2}) \otimes p_{2}^{*} O_{S_{2}} (D_{2})

, with

D_{2} = n_{2} H_{Z}

. Then, we can construct a code on

T_{2}

over

F_{q^{2}}

with parameters

1.: $n = (q^{5} + 1) (q^{3} + 1) {(q^{2} + 1)}^{2}$ ,
2.: $k = h^{0} (S_{2}, O_{S_{2}} (n_{2} H_{Z})) + \sum_{t = 1}^{b_{2}} h^{0} (S_{2}, O_{S_{2}} ((n_{2} - n_{1} t) H_{Z}))$ ,
3.: $d \geq (q^{5} + 1) (q^{3} + 1) {(q^{2} + 1)}^{2} - (q^{5} + 1) (q^{3} + 1) (q^{2} + 1) b_{2} - (q^{5} + 1) (q^{3} + 1) (2 b_{2} n_{1} + n_{2}) (q^{2} + 1)$ .

Proof.

Let

T_{2, B_{j}} = P (O_{B_{j}} \oplus O_{B_{j}} (- n_{1}))

. Now, let us consider the restriction

{L |}_{T_{2, B_{j}}} = O_{T_{2, B_{j}}} (b_{2}) \otimes p_{2} |_{T_{2, B_{j}}}^{*} O_{B_{j}} (n_{2})

. Moreover, let us consider the divisor

H_{2} = ς + 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}}} \leq \frac{H_{2} {\cdot L |}_{T_{2, B_{j}}}}{m i n \{H_{2} \cdot f\}} = 2 b_{2} n_{1} + n_{2},

(52)

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

d \geq (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) (2 b_{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.: $d \geq 12$ .

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.: $d \geq 1485$ .

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.: $d \geq 92$ .

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.

Reference

Camazón Portela, D.; López Ramos, J.A. Error-Correcting Codes on Projective Bundles over Deligne-Lusztig Varieties. Mathematics 2023, 11, 3079. [Google Scholar] [CrossRef]

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

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