MinT - Best Known (76−53, 76, s)-Nets in Base 128

Best Known (76−53, 76, s)-Nets in Base 128

(76−53, 76, 288)-Net over F₁₂₈ — Constructive and digital

Digital (23, 76, 288)-net over F₁₂₈, using

t-expansion [i] based on digital (9, 76, 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]

(76−53, 76, 386)-Net over F₁₂₈ — Digital

Digital (23, 76, 386)-net over F₁₂₈, using

t-expansion [i] based on digital (15, 76, 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]

(76−53, 76, 513)-Net in Base 128

(23, 76, 513)-net in base 128, using

128⁴ times duplication [i] based on (19, 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]

(76−53, 76, 99523)-Net in Base 128 — Upper bound on s

There is no (23, 76, 99524)-net in base 128, because

1 times m-reduction [i] would yield (23, 75, 99524)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 109 860712 864395 036409 078772 253945 545713 104920 942853 242148 554304 352933 348234 559375 655452 663419 153171 457530 514595 578549 370317 994233 561964 370346 277071 924477 257832 > 128⁷⁵ [i]