MinT - Best Known (73−40, 73, s)-Nets in Base 128

Best Known (73−40, 73, s)-Nets in Base 128

(73−40, 73, 480)-Net over F₁₂₈ — Constructive and digital

Digital (33, 73, 480)-net over F₁₂₈, using

2 times m-reduction [i] based on digital (33, 75, 480)-net over F₁₂₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (3, 24, 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, 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]

(73−40, 73, 546)-Net in Base 128 — Constructive

(33, 73, 546)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (4, 24, 258)-net in base 128, using
  - base change [i] based on digital (1, 21, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]
2. digital (9, 49, 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]

(73−40, 73, 1086)-Net over F₁₂₈ — Digital

Digital (33, 73, 1086)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(73−40, 73, 3212311)-Net in Base 128 — Upper bound on s

There is no (33, 73, 3212312)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 6703 943059 024585 040310 611002 462415 869088 975969 831925 552069 265523 736953 889513 840483 978739 630726 105349 246036 743180 141015 334770 452067 151390 579378 850530 297899 > 128⁷³ [i]