MinT - Best Known (78−45, 78, s)-Nets in Base 128

Best Known (78−45, 78, s)-Nets in Base 128

(78−45, 78, 438)-Net over F₁₂₈ — Constructive and digital

Digital (33, 78, 438)-net over F₁₂₈, using

128¹ times duplication [i] based on digital (32, 77, 438)-net over F₁₂₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (1, 23, 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, 54, 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]

(78−45, 78, 514)-Net in Base 128 — Constructive

(33, 78, 514)-net in base 128, using

(u,Â u+v)-construction [i] based on
1. (4, 26, 257)-net in base 128, using
  - 6 times m-reduction [i] based on (4, 32, 257)-net in base 128, using
    - base change [i] based on digital (0, 28, 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]
2. (7, 52, 257)-net in base 128, using
  - 4 times m-reduction [i] based on (7, 56, 257)-net in base 128, using
    - base change [i] based on digital (0, 49, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(78−45, 78, 761)-Net over F₁₂₈ — Digital

Digital (33, 78, 761)-net over F₁₂₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(78−45, 78, 1691536)-Net in Base 128 — Upper bound on s

There is no (33, 78, 1691537)-net in base 128, because

1 times m-reduction [i] would yield (33, 77, 1691537)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 1 799569 379511 124514 685749 405012 064215 502147 603219 401077 005603 076568 684810 827623 694694 519757 256493 121111 416652 012453 343429 573119 593837 363719 540650 164109 310450 113984 > 128⁷⁷ [i]