MinT - Best Known (116−79, 116, s)-Nets in Base 16

Best Known (116−79, 116, s)-Nets in Base 16

(116−79, 116, 65)-Net over F₁₆ — Constructive and digital

Digital (37, 116, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 116, 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]

(116−79, 116, 120)-Net in Base 16 — Constructive

(37, 116, 120)-net in base 16, using

14 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]

(116−79, 116, 208)-Net over F₁₆ — Digital

Digital (37, 116, 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]

(116−79, 116, 3625)-Net in Base 16 — Upper bound on s

There is no (37, 116, 3626)-net in base 16, because

1 times m-reduction [i] would yield (37, 115, 3626)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 993006 434714 017267 743455 100355 366390 533178 105607 496028 821363 071540 041854 210161 353998 124751 483984 060600 325696 682796 738698 998667 995954 151436 > 16¹¹⁵ [i]