MinT - Best Known (37, 128, s)-Nets in Base 16

Best Known (37, 128, s)-Nets in Base 16

(37, 128, 65)-Net over F₁₆ — Constructive and digital

Digital (37, 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]

(37, 128, 120)-Net in Base 16 — Constructive

(37, 128, 120)-net in base 16, using

2 times m-reduction [i] based on (37, 130, 120)-net in base 16, using
- base change [i] based on digital (11, 104, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
      - a function field by SÃ©mirat [i]

(37, 128, 208)-Net over F₁₆ — Digital

Digital (37, 128, 208)-net over F₁₆, using

net from sequence [i] based on digital (37, 207)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 37 and N(F) ≥ 208, using
  - Drinfeld modules of rank 1 [i]

(37, 128, 2915)-Net in Base 16 — Upper bound on s

There is no (37, 128, 2916)-net in base 16, because

1 times m-reduction [i] would yield (37, 127, 2916)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 847 660891 216616 812321 300466 373690 545932 619744 700764 386804 387228 707023 417052 044484 786258 812886 570313 506916 623968 731687 298878 029855 679291 630028 657465 444176 > 16¹²⁷ [i]