MinT - Best Known (13, 13+21, s)-Nets in Base 128

Best Known (13, 13+21, s)-Nets in Base 128

(13, 13+21, 321)-Net over F₁₂₈ — Constructive and digital

Digital (13, 34, 321)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 10, 129)-net over F₁₂₈, using
  - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
      - the rational function field F₁₂₈(x) [i]
    - Niederreiter sequence [i]
2. digital (3, 24, 192)-net over F₁₂₈, using
  - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
      - a function field by SÃ©mirat [i]

(13, 13+21, 322)-Net over F₁₂₈ — Digital

Digital (13, 34, 322)-net over F₁₂₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(128³⁴, 322, F₁₂₈, 2, 21) (dual of [(322, 2), 610, 22]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(128¹¹, 150, F₁₂₈, 2, 10) (dual of [(150, 2), 289, 11]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,289P) [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
      - a function field by SÃ©mirat [i]
  2. linear OOA(128²³, 172, F₁₂₈, 2, 21) (dual of [(172, 2), 321, 22]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,322P) [i] based on function field F/F₁₂₈ with g(F) = 2 and N(F) ≥ 172, using
      - sharp bound on the number of rational points on curves with genus 2 [i]

(13, 13+21, 386)-Net in Base 128 — Constructive

(13, 34, 386)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. digital (0, 10, 129)-net over F₁₂₈, using
  - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
      - the rational function field F₁₂₈(x) [i]
    - Niederreiter sequence [i]
2. (3, 24, 257)-net in base 128, using
  - base change [i] based on digital (0, 21, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(13, 13+21, 513)-Net in Base 128

(13, 34, 513)-net in base 128, using

6 times m-reduction [i] based on (13, 40, 513)-net in base 128, using
- base change [i] based on digital (8, 35, 513)-net over F₂₅₆, using
  - net from sequence [i] based on digital (8, 512)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 8 and N(F) ≥ 513, using
      - K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₂₅₆ [i]

(13, 13+21, 320596)-Net in Base 128 — Upper bound on s

There is no (13, 34, 320597)-net in base 128, because

1 times m-reduction [i] would yield (13, 33, 320597)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 3450 887033 787418 362934 692413 382579 245913 500491 082353 188248 554325 558916 > 128³³ [i]