MinT - Best Known (36, 78, s)-Nets in Base 128

Best Known (36, 78, s)-Nets in Base 128

(36, 78, 504)-Net over F₁₂₈ — Constructive and digital

Digital (36, 78, 504)-net over F₁₂₈, using

t-expansion [i] based on digital (35, 78, 504)-net over F₁₂₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 26, 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]
  2. digital (9, 52, 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]

(36, 78, 547)-Net in Base 128 — Constructive

(36, 78, 547)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (6, 27, 259)-net in base 128, using
  - 5 times m-reduction [i] based on (6, 32, 259)-net in base 128, using
    - base change [i] based on digital (2, 28, 259)-net over F₂₅₆, using
      - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
        Niederreiter sequence [i]
2. digital (9, 51, 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]

(36, 78, 1318)-Net over F₁₂₈ — Digital

Digital (36, 78, 1318)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(36, 78, 4586356)-Net in Base 128 — Upper bound on s

There is no (36, 78, 4586357)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 230 345344 787877 506778 606215 607233 019531 643880 053672 082215 568191 566250 779760 182549 977979 740773 283856 222452 679292 040339 929400 450205 247917 702685 747641 860316 166015 985120 > 128⁷⁸ [i]