MinT - Best Known (49−26, 49, s)-Nets in Base 128

Best Known (49−26, 49, s)-Nets in Base 128

(49−26, 49, 438)-Net over F₁₂₈ — Constructive and digital

Digital (23, 49, 438)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 14, 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, 35, 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]

(49−26, 49, 517)-Net in Base 128 — Constructive

(23, 49, 517)-net in base 128, using

1 times m-reduction [i] based on (23, 50, 517)-net in base 128, using
- (u,Â u+v)-construction [i] based on
  1. (3, 16, 258)-net in base 128, using
    - base change [i] based on digital (1, 14, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]
  2. (7, 34, 259)-net in base 128, using
    - 6 times m-reduction [i] based on (7, 40, 259)-net in base 128, using
      - base change [i] based on digital (2, 35, 259)-net over F₂₅₆, using
        net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(49−26, 49, 1094)-Net over F₁₂₈ — Digital

Digital (23, 49, 1094)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(49−26, 49, 3909871)-Net in Base 128 — Upper bound on s

There is no (23, 49, 3909872)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 17 918014 588526 051827 525534 010256 653378 339624 683348 899209 776958 532006 931102 796810 105119 672419 353841 396171 > 128⁴⁹ [i]