MinT - Best Known (15, 15+24, s)-Nets in Base 128

Best Known (15, 15+24, s)-Nets in Base 128

(15, 15+24, 321)-Net over F₁₂₈ — Constructive and digital

Digital (15, 39, 321)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 12, 129)-net over F₁₂₈, using
  - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
      - the rational function field F₁₂₈(x) [i]
    - Niederreiter sequence [i]
2. digital (3, 27, 192)-net over F₁₂₈, using
  - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
      - a function field by SÃ©mirat [i]

(15, 15+24, 386)-Net over F₁₂₈ — Digital

Digital (15, 39, 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]

(15, 15+24, 513)-Net in Base 128

(15, 39, 513)-net in base 128, using

17 times m-reduction [i] based on (15, 56, 513)-net in base 128, using
- base change [i] based on digital (8, 49, 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]

(15, 15+24, 293752)-Net in Base 128 — Upper bound on s

There is no (15, 39, 293753)-net in base 128, because

the generalized Rao bound for nets shows that 128^mÂ â‰¥ 15177 305160 630092 638635 467680 387287 700292 405415 192276 424899 465837 786152 850435 360308 > 128³⁹ [i]