MinT - Best Known (26, 26+39, s)-Nets in Base 128

Best Known (26, 26+39, s)-Nets in Base 128

(26, 26+39, 384)-Net over F₁₂₈ — Constructive and digital

Digital (26, 65, 384)-net over F₁₂₈, using

1 times m-reduction [i] based on digital (26, 66, 384)-net over F₁₂₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (3, 23, 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 (3, 43, 192)-net over F₁₂₈, using
    - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈ (see above)

(26, 26+39, 407)-Net in Base 128 — Constructive

(26, 65, 407)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. digital (1, 20, 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. (6, 45, 257)-net in base 128, using
  - 3 times m-reduction [i] based on (6, 48, 257)-net in base 128, using
    - base change [i] based on digital (0, 42, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]

(26, 26+39, 514)-Net over F₁₂₈ — Digital

Digital (26, 65, 514)-net over F₁₂₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(128⁶⁵, 514, F₁₂₈, 2, 39) (dual of [(514, 2), 963, 40]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(128⁶³, 513, F₁₂₈, 2, 39) (dual of [(513, 2), 963, 40]-NRT-code), using
  - extended algebraic-geometric NRT-code AG_e(2;F,986P) [i] based on function field F/F₁₂₈ with g(F) = 24 and N(F) ≥ 513, using
    - a curve from the manYPoints database [i]

(26, 26+39, 782306)-Net in Base 128 — Upper bound on s

There is no (26, 65, 782307)-net in base 128, because

1 times m-reduction [i] would yield (26, 64, 782307)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 726 841462 417498 178848 432582 293205 075185 650237 963108 651964 793715 464870 756385 376470 402661 286162 142540 164400 250836 266200 716324 503957 911712 > 128⁶⁴ [i]