MinT - Best Known (65−37, 65, s)-Nets in Base 128

Best Known (65−37, 65, s)-Nets in Base 128

(65−37, 65, 438)-Net over F₁₂₈ — Constructive and digital

Digital (28, 65, 438)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 19, 150)-net over F₁₂₈, using
  - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
      - a function field by SÃ©mirat [i]
2. digital (9, 46, 288)-net over F₁₂₈, using
  - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
      - a function field by SÃ©mirat [i]

(65−37, 65, 515)-Net in Base 128 — Constructive

(28, 65, 515)-net in base 128, using

128¹ times duplication [i] based on (27, 64, 515)-net in base 128, using
- base change [i] based on digital (19, 56, 515)-net over F₂₅₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 18, 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]
    2. digital (1, 38, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(65−37, 65, 735)-Net over F₁₂₈ — Digital

Digital (28, 65, 735)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(65−37, 65, 1847678)-Net in Base 128 — Upper bound on s

There is no (28, 65, 1847679)-net in base 128, because

1 times m-reduction [i] would yield (28, 64, 1847679)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 726 841857 546235 830046 343326 933584 011752 099672 778381 795895 313387 908783 749496 207348 224764 734956 913877 977768 228557 989536 832762 035599 199995 > 128⁶⁴ [i]