MinT - Best Known (123−91, 123, s)-Nets in Base 16

Best Known (123−91, 123, s)-Nets in Base 16

(123−91, 123, 65)-Net over F₁₆ — Constructive and digital

Digital (32, 123, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 123, 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]

(123−91, 123, 98)-Net in Base 16 — Constructive

(32, 123, 98)-net in base 16, using

2 times m-reduction [i] based on (32, 125, 98)-net in base 16, using
- base change [i] based on digital (7, 100, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
      - a function field by SÃ©mirat [i]

(123−91, 123, 168)-Net over F₁₆ — Digital

Digital (32, 123, 168)-net over F₁₆, using

t-expansion [i] based on digital (31, 123, 168)-net over F₁₆, using
- net from sequence [i] based on digital (31, 167)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 31 and N(F) ≥ 168, using
    - a curve from the manYPoints database [i]

(123−91, 123, 2135)-Net in Base 16 — Upper bound on s

There is no (32, 123, 2136)-net in base 16, because

1 times m-reduction [i] would yield (32, 122, 2136)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 803 736922 536154 303802 806493 113525 329845 985663 321028 483018 646337 362158 194279 912922 205294 506307 048555 681924 356132 848058 670944 305892 066695 579659 712676 > 16¹²² [i]