MinT - Best Known (69−53, 69, s)-Nets in Base 128

Best Known (69−53, 69, s)-Nets in Base 128

(69−53, 69, 288)-Net over F₁₂₈ — Constructive and digital

Digital (16, 69, 288)-net over F₁₂₈, using

t-expansion [i] based on digital (9, 69, 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]

(69−53, 69, 386)-Net over F₁₂₈ — Digital

Digital (16, 69, 386)-net over F₁₂₈, using

t-expansion [i] based on digital (15, 69, 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]

(69−53, 69, 26943)-Net in Base 128 — Upper bound on s

There is no (16, 69, 26944)-net in base 128, because

1 times m-reduction [i] would yield (16, 68, 26944)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 195260 357803 609805 234606 340859 586175 068177 085722 513312 428221 291760 371465 169897 755179 241488 649242 126739 613146 585636 519408 782513 860631 662709 461213 > 128⁶⁸ [i]