MinT - Best Known (35, 35+81, s)-Nets in Base 16

Best Known (35, 35+81, s)-Nets in Base 16

(35, 35+81, 65)-Net over F₁₆ — Constructive and digital

Digital (35, 116, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 116, 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]

(35, 35+81, 120)-Net in Base 16 — Constructive

(35, 116, 120)-net in base 16, using

4 times m-reduction [i] based on (35, 120, 120)-net in base 16, using
- base change [i] based on digital (11, 96, 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]

(35, 35+81, 193)-Net over F₁₆ — Digital

Digital (35, 116, 193)-net over F₁₆, using

t-expansion [i] based on digital (33, 116, 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]

(35, 35+81, 3022)-Net in Base 16 — Upper bound on s

There is no (35, 116, 3023)-net in base 16, because

1 times m-reduction [i] would yield (35, 115, 3023)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 993779 038110 408589 088714 469452 857323 072218 860474 516832 718796 182284 229137 999758 392008 686809 633924 080263 144163 201224 063845 740271 310201 329926 > 16¹¹⁵ [i]