MinT - Best Known (29, 29+57, s)-Nets in Base 16

Best Known (29, 29+57, s)-Nets in Base 16

(29, 29+57, 65)-Net over F₁₆ — Constructive and digital

Digital (29, 86, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 86, 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+57, 120)-Net in Base 16 — Constructive

(29, 86, 120)-net in base 16, using

4 times m-reduction [i] based on (29, 90, 120)-net in base 16, using
- base change [i] based on digital (11, 72, 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]

(29, 29+57, 161)-Net over F₁₆ — Digital

Digital (29, 86, 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+57, 3390)-Net in Base 16 — Upper bound on s

There is no (29, 86, 3391)-net in base 16, because

1 times m-reduction [i] would yield (29, 85, 3391)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 241357 799025 564662 805514 091560 334293 762397 241621 101133 947783 421003 801797 502963 284588 725178 654501 082521 > 16⁸⁵ [i]