MinT - Best Known (78−46, 78, s)-Nets in Base 128

Best Known (78−46, 78, s)-Nets in Base 128

(78−46, 78, 417)-Net over F₁₂₈ — Constructive and digital

Digital (32, 78, 417)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 23, 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 (9, 55, 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]

(78−46, 78, 432)-Net in Base 128 — Constructive

(32, 78, 432)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (4, 27, 257)-net in base 128, using
  - 5 times m-reduction [i] based on (4, 32, 257)-net in base 128, using
    - base change [i] based on digital (0, 28, 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 (5, 51, 216)-net over F₁₂₈, using
  - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
      - a function field by SÃ©mirat [i]

(78−46, 78, 643)-Net over F₁₂₈ — Digital

Digital (32, 78, 643)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(78−46, 78, 1039532)-Net in Base 128 — Upper bound on s

There is no (32, 78, 1039533)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 230 345134 961375 286667 454258 662787 979999 697247 141069 525351 065319 809470 254217 227176 098968 091990 013594 009285 977955 993519 239671 131728 150524 664593 797479 703200 503109 511152 > 128⁷⁸ [i]