MinT - Best Known (25, 25+31, s)-Nets in Base 128

Best Known (25, 25+31, s)-Nets in Base 128

(25, 25+31, 438)-Net over F₁₂₈ — Constructive and digital

Digital (25, 56, 438)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 16, 150)-net over F₁₂₈, using
  - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
      - a function field by SÃ©mirat [i]
2. digital (9, 40, 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]

(25, 25+31, 517)-Net in Base 128 — Constructive

(25, 56, 517)-net in base 128, using

base change [i] based on digital (18, 49, 517)-net over F₂₅₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (1, 16, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]
  2. digital (2, 33, 259)-net over F₂₅₆, using
    - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(25, 25+31, 829)-Net over F₁₂₈ — Digital

Digital (25, 56, 829)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(25, 25+31, 2694004)-Net in Base 128 — Upper bound on s

There is no (25, 56, 2694005)-net in base 128, because

1 times m-reduction [i] would yield (25, 55, 2694005)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 78 804202 259237 062958 036826 286886 346623 569388 497827 465692 481687 434838 622236 310211 919452 153119 419557 503631 302494 799312 > 128⁵⁵ [i]