MinT - Best Known (35, 35+42, s)-Nets in Base 128

Best Known (35, 35+42, s)-Nets in Base 128

(35, 35+42, 504)-Net over F₁₂₈ — Constructive and digital

Digital (35, 77, 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, 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]

(35, 35+42, 546)-Net in Base 128 — Constructive

(35, 77, 546)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (5, 26, 258)-net in base 128, using
  - 6 times m-reduction [i] based on (5, 32, 258)-net in base 128, using
    - base change [i] based on digital (1, 28, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-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]

(35, 35+42, 1173)-Net over F₁₂₈ — Digital

Digital (35, 77, 1173)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(35, 35+42, 3640191)-Net in Base 128 — Upper bound on s

There is no (35, 77, 3640192)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 1 799574 395774 990217 437655 786397 173759 436903 317640 198592 062100 200698 064671 023270 730919 378089 258532 454201 622144 812704 959625 083232 423285 361826 898432 370138 934160 089145 > 128⁷⁷ [i]