MinT - Best Known (29, 72, s)-Nets in Base 16

Best Known (29, 72, s)-Nets in Base 16

(29, 72, 98)-Net over F₁₆ — Constructive and digital

Digital (29, 72, 98)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 23, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
      - a function field by SÃ©mirat [i]
2. digital (6, 49, 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, 72, 128)-Net in Base 16 — Constructive

(29, 72, 128)-net in base 16, using

base change [i] based on digital (5, 48, 128)-net over F₆₄, using
- net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
    - a function field by SÃ©mirat [i]

(29, 72, 161)-Net over F₁₆ — Digital

Digital (29, 72, 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, 72, 6803)-Net in Base 16 — Upper bound on s

There is no (29, 72, 6804)-net in base 16, because

1 times m-reduction [i] would yield (29, 71, 6804)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 31 108878 043863 818681 679253 467043 585089 374796 454624 775791 411541 681854 607824 372768 575136 > 16⁷¹ [i]