MinT - Best Known (108−79, 108, s)-Nets in Base 16

Best Known (108−79, 108, s)-Nets in Base 16

(108−79, 108, 65)-Net over F₁₆ — Constructive and digital

Digital (29, 108, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 108, 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]

(108−79, 108, 98)-Net in Base 16 — Constructive

(29, 108, 98)-net in base 16, using

2 times m-reduction [i] based on (29, 110, 98)-net in base 16, using
- base change [i] based on digital (7, 88, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
      - a function field by SÃ©mirat [i]

(108−79, 108, 161)-Net over F₁₆ — Digital

Digital (29, 108, 161)-net over F₁₆, using

net from sequence [i] based on digital (29, 160)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 29 and N(F) ≥ 161, using
  - a curve from the manYPoints database [i]

(108−79, 108, 2043)-Net in Base 16 — Upper bound on s

There is no (29, 108, 2044)-net in base 16, because

1 times m-reduction [i] would yield (29, 107, 2044)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 698 743196 771254 234521 485215 295702 402331 516107 761633 420123 879379 475442 925874 510904 025966 827537 539701 215160 160776 490013 005505 742866 > 16¹⁰⁷ [i]