MinT - Best Known (73−54, 73, s)-Nets in Base 128

Best Known (73−54, 73, s)-Nets in Base 128

(73−54, 73, 288)-Net over F₁₂₈ — Constructive and digital

Digital (19, 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]

(73−54, 73, 386)-Net over F₁₂₈ — Digital

Digital (19, 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]

(73−54, 73, 513)-Net in Base 128

(19, 73, 513)-net in base 128, using

128¹ times duplication [i] based on (18, 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]

(73−54, 73, 42829)-Net in Base 128 — Upper bound on s

There is no (19, 73, 42830)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 6707 060827 201876 653535 660950 720366 842533 269306 703022 068159 474993 583118 674970 972253 288212 494901 938583 768869 553835 555529 516091 160270 963175 479060 830314 427304 > 128⁷³ [i]