MinT - Best Known (18, 51, s)-Nets in Base 128

Best Known (18, 51, s)-Nets in Base 128

(18, 51, 300)-Net over F₁₂₈ — Constructive and digital

Digital (18, 51, 300)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 17, 150)-net over F₁₂₈, using
  - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
      - a function field by SÃ©mirat [i]
2. digital (1, 34, 150)-net over F₁₂₈, using
  - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈ (see above)

(18, 51, 386)-Net over F₁₂₈ — Digital

Digital (18, 51, 386)-net over F₁₂₈, using

t-expansion [i] based on digital (15, 51, 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]

(18, 51, 513)-Net in Base 128

(18, 51, 513)-net in base 128, using

t-expansion [i] based on (17, 51, 513)-net in base 128, using
- 21 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]

(18, 51, 205944)-Net in Base 128 — Upper bound on s

There is no (18, 51, 205945)-net in base 128, because

1 times m-reduction [i] would yield (18, 50, 205945)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 2293 597887 397324 541007 027116 371706 951414 537244 306651 425239 963320 286102 203772 688931 587518 104618 009915 480045 > 128⁵⁰ [i]