MinT - Best Known (123−89, 123, s)-Nets in Base 16

Best Known (123−89, 123, s)-Nets in Base 16

(123−89, 123, 65)-Net over F₁₆ — Constructive and digital

Digital (34, 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]

(123−89, 123, 104)-Net in Base 16 — Constructive

(34, 123, 104)-net in base 16, using

2 times m-reduction [i] based on (34, 125, 104)-net in base 16, using
- base change [i] based on digital (9, 100, 104)-net over F₃₂, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(123−89, 123, 193)-Net over F₁₆ — Digital

Digital (34, 123, 193)-net over F₁₆, using

t-expansion [i] based on digital (33, 123, 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]

(123−89, 123, 2484)-Net in Base 16 — Upper bound on s

There is no (34, 123, 2485)-net in base 16, because

1 times m-reduction [i] would yield (34, 122, 2485)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 802 227742 834534 280837 187690 221813 764426 452931 206773 949754 715904 411816 780405 640235 437276 533631 780186 534664 066951 601892 529524 834762 122584 407565 143726 > 16¹²² [i]