MinT - Best Known (12, 31, s)-Nets in Base 128

Best Known (12, 31, s)-Nets in Base 128

(12, 31, 321)-Net over F₁₂₈ — Constructive and digital

Digital (12, 31, 321)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 9, 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, 22, 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]

(12, 31, 322)-Net over F₁₂₈ — Digital

Digital (12, 31, 322)-net over F₁₂₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(128³¹, 322, F₁₂₈, 2, 19) (dual of [(322, 2), 613, 20]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(128¹⁰, 150, F₁₂₈, 2, 9) (dual of [(150, 2), 290, 10]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,290P) [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(128²¹, 172, F₁₂₈, 2, 19) (dual of [(172, 2), 323, 20]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,324P) [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]

(12, 31, 386)-Net in Base 128 — Constructive

(12, 31, 386)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. digital (0, 9, 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, 22, 257)-net in base 128, using
  - 2 times m-reduction [i] based on (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]

(12, 31, 513)-Net in Base 128

(12, 31, 513)-net in base 128, using

1 times m-reduction [i] based on (12, 32, 513)-net in base 128, using
- base change [i] based on digital (8, 28, 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]

(12, 31, 345124)-Net in Base 128 — Upper bound on s

There is no (12, 31, 345125)-net in base 128, because

1 times m-reduction [i] would yield (12, 30, 345125)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 1645 524436 279789 081042 426942 489338 739321 248596 057512 692931 156176 > 128³⁰ [i]