MinT - Best Known (34, 109, s)-Nets in Base 16

Best Known (34, 109, s)-Nets in Base 16

(34, 109, 65)-Net over F₁₆ — Constructive and digital

Digital (34, 109, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 109, 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]

(34, 109, 120)-Net in Base 16 — Constructive

(34, 109, 120)-net in base 16, using

6 times m-reduction [i] based on (34, 115, 120)-net in base 16, using
- base change [i] based on digital (11, 92, 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]

(34, 109, 193)-Net over F₁₆ — Digital

Digital (34, 109, 193)-net over F₁₆, using

t-expansion [i] based on digital (33, 109, 193)-net over F₁₆, using
- net from sequence [i] based on digital (33, 192)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 33 and N(F) ≥ 193, using
    - a curve from the manYPoints database [i]

(34, 109, 3175)-Net in Base 16 — Upper bound on s

There is no (34, 109, 3176)-net in base 16, because

1 times m-reduction [i] would yield (34, 108, 3176)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 11201 406158 391187 387157 081889 074774 102426 350345 477840 478388 099220 439344 070775 010219 088052 888614 472195 187398 949427 323566 998862 843781 > 16¹⁰⁸ [i]