MinT - Best Known (29, 29+69, s)-Nets in Base 16

Best Known (29, 29+69, s)-Nets in Base 16

(29, 29+69, 65)-Net over F₁₆ — Constructive and digital

Digital (29, 98, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 98, 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]

(29, 29+69, 104)-Net in Base 16 — Constructive

(29, 98, 104)-net in base 16, using

2 times m-reduction [i] based on (29, 100, 104)-net in base 16, using
- base change [i] based on digital (9, 80, 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]

(29, 29+69, 161)-Net over F₁₆ — Digital

Digital (29, 98, 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]

(29, 29+69, 2439)-Net in Base 16 — Upper bound on s

There is no (29, 98, 2440)-net in base 16, because

1 times m-reduction [i] would yield (29, 97, 2440)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 632 575640 747704 298521 206135 841223 348449 263022 948602 290685 804357 279586 240819 583470 202710 482301 950586 669702 771704 358276 > 16⁹⁷ [i]