MinT - Best Known (77−54, 77, s)-Nets in Base 128

Best Known (77−54, 77, s)-Nets in Base 128

(77−54, 77, 288)-Net over F₁₂₈ — Constructive and digital

Digital (23, 77, 288)-net over F₁₂₈, using

t-expansion [i] based on digital (9, 77, 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]

(77−54, 77, 386)-Net over F₁₂₈ — Digital

Digital (23, 77, 386)-net over F₁₂₈, using

t-expansion [i] based on digital (15, 77, 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]

(77−54, 77, 513)-Net in Base 128

(23, 77, 513)-net in base 128, using

128⁵ 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]

(77−54, 77, 87899)-Net in Base 128 — Upper bound on s

There is no (23, 77, 87900)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 1 799615 368964 118366 776724 424119 912350 698547 797201 607489 437792 952351 362197 407662 416694 287440 498963 497086 901582 763306 541673 386317 908450 142731 069570 525788 979185 247163 > 128⁷⁷ [i]