MinT - Best Known (28−17, 28, s)-Nets in Base 128

Best Known (28−17, 28, s)-Nets in Base 128

(28−17, 28, 321)-Net over F₁₂₈ — Constructive and digital

Digital (11, 28, 321)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 8, 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, 20, 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]

(28−17, 28, 322)-Net over F₁₂₈ — Digital

Digital (11, 28, 322)-net over F₁₂₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(128²⁸, 322, F₁₂₈, 2, 17) (dual of [(322, 2), 616, 18]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(128⁹, 150, F₁₂₈, 2, 8) (dual of [(150, 2), 291, 9]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,291P) [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(128¹⁹, 172, F₁₂₈, 2, 17) (dual of [(172, 2), 325, 18]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,326P) [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]

(28−17, 28, 386)-Net in Base 128 — Constructive

(11, 28, 386)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. digital (0, 8, 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, 20, 257)-net in base 128, using
  - 4 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]

(28−17, 28, 383456)-Net in Base 128 — Upper bound on s

There is no (11, 28, 383457)-net in base 128, because

1 times m-reduction [i] would yield (11, 27, 383457)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 784 648029 821317 673956 691235 976303 475402 558774 760700 233700 > 128²⁷ [i]