MinT - Best Known (128−93, 128, s)-Nets in Base 16

Best Known (128−93, 128, s)-Nets in Base 16

(128−93, 128, 65)-Net over F₁₆ — Constructive and digital

Digital (35, 128, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 128, 65)-net over F₁₆, using
- net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
    - the Hermitian function field over F₁₆ [i]

(128−93, 128, 104)-Net in Base 16 — Constructive

(35, 128, 104)-net in base 16, using

2 times m-reduction [i] based on (35, 130, 104)-net in base 16, using
- base change [i] based on digital (9, 104, 104)-net over F₃₂, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(128−93, 128, 193)-Net over F₁₆ — Digital

Digital (35, 128, 193)-net over F₁₆, using

t-expansion [i] based on digital (33, 128, 193)-net over F₁₆, using
- net from sequence [i] based on digital (33, 192)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 33 and N(F) ≥ 193, using
    - a curve from the manYPoints database [i]

(128−93, 128, 2507)-Net in Base 16 — Upper bound on s

There is no (35, 128, 2508)-net in base 16, because

1 times m-reduction [i] would yield (35, 127, 2508)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 852 734516 670599 386066 595756 741779 456733 603121 650042 290308 052240 851417 182907 068403 488062 210624 775043 645577 668666 140666 529788 360569 763880 295312 716745 010896 > 16¹²⁷ [i]