MinT - Best Known (67−47, 67, s)-Nets in Base 128

Best Known (67−47, 67, s)-Nets in Base 128

(67−47, 67, 288)-Net over F₁₂₈ — Constructive and digital

Digital (20, 67, 288)-net over F₁₂₈, using

t-expansion [i] based on digital (9, 67, 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]

(67−47, 67, 386)-Net over F₁₂₈ — Digital

Digital (20, 67, 386)-net over F₁₂₈, using

t-expansion [i] based on digital (15, 67, 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]

(67−47, 67, 513)-Net in Base 128

(20, 67, 513)-net in base 128, using

t-expansion [i] based on (17, 67, 513)-net in base 128, using
- 5 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]

(67−47, 67, 82674)-Net in Base 128 — Upper bound on s

There is no (20, 67, 82675)-net in base 128, because

1 times m-reduction [i] would yield (20, 66, 82675)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 11 911010 503495 659618 549951 805099 605306 204201 112332 930287 657366 483083 096972 431474 519620 672851 852968 326703 367748 023658 173453 370808 648225 925856 > 128⁶⁶ [i]