MinT - Best Known (78−44, 78, s)-Nets in Base 128

Best Known (78−44, 78, s)-Nets in Base 128

(78−44, 78, 480)-Net over F₁₂₈ — Constructive and digital

Digital (34, 78, 480)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 25, 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]
2. digital (9, 53, 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]

(78−44, 78, 515)-Net in Base 128 — Constructive

(34, 78, 515)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (4, 26, 257)-net in base 128, using
  - 6 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. (8, 52, 258)-net in base 128, using
  - 4 times m-reduction [i] based on (8, 56, 258)-net in base 128, using
    - base change [i] based on digital (1, 49, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(78−44, 78, 905)-Net over F₁₂₈ — Digital

Digital (34, 78, 905)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(78−44, 78, 2108940)-Net in Base 128 — Upper bound on s

There is no (34, 78, 2108941)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 230 346216 795171 090383 970344 815864 584828 858361 548759 183397 265675 221598 351323 643655 350703 829641 915083 333123 359810 628433 174270 229699 222566 567210 198676 826967 938433 095552 > 128⁷⁸ [i]