MinT - Best Known (68−49, 68, s)-Nets in Base 128

Best Known (68−49, 68, s)-Nets in Base 128

(68−49, 68, 288)-Net over F₁₂₈ — Constructive and digital

Digital (19, 68, 288)-net over F₁₂₈, using

t-expansion [i] based on digital (9, 68, 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]

(68−49, 68, 386)-Net over F₁₂₈ — Digital

Digital (19, 68, 386)-net over F₁₂₈, using

t-expansion [i] based on digital (15, 68, 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]

(68−49, 68, 513)-Net in Base 128

(19, 68, 513)-net in base 128, using

t-expansion [i] based on (17, 68, 513)-net in base 128, using
- 4 times m-reduction [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]

(68−49, 68, 58898)-Net in Base 128 — Upper bound on s

There is no (19, 68, 58899)-net in base 128, because

1 times m-reduction [i] would yield (19, 67, 58899)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 1524 902235 886382 978054 807757 015243 549286 577540 766297 847897 550751 976362 841109 233379 166672 478309 893026 555947 993149 482054 615182 000253 014798 221248 > 128⁶⁷ [i]