MinT - Best Known (73−51, 73, s)-Nets in Base 128

Best Known (73−51, 73, s)-Nets in Base 128

(73−51, 73, 288)-Net over F₁₂₈ — Constructive and digital

Digital (22, 73, 288)-net over F₁₂₈, using

t-expansion [i] based on digital (9, 73, 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]

(73−51, 73, 386)-Net over F₁₂₈ — Digital

Digital (22, 73, 386)-net over F₁₂₈, using

t-expansion [i] based on digital (15, 73, 386)-net over F₁₂₈, using
- net from sequence [i] based on digital (15, 385)-sequence over F₁₂₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 15 and N(F) ≥ 386, using
    - a curve from the manYPoints database [i]

(73−51, 73, 513)-Net in Base 128

(22, 73, 513)-net in base 128, using

128¹ times duplication [i] based on (21, 72, 513)-net in base 128, using
- t-expansion [i] based on (17, 72, 513)-net in base 128, using
  - base change [i] based on digital (8, 63, 513)-net over F₂₅₆, using
    - net from sequence [i] based on digital (8, 512)-sequence over F₂₅₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 8 and N(F) ≥ 513, using
        K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₂₅₆ [i]

(73−51, 73, 93870)-Net in Base 128 — Upper bound on s

There is no (22, 73, 93871)-net in base 128, because

1 times m-reduction [i] would yield (22, 72, 93871)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 52 377413 402250 303051 571940 844183 682997 128329 650574 158817 829688 591106 421214 155139 089854 538047 180986 155880 169554 736432 329124 052583 333854 459716 961792 923836 > 128⁷² [i]