MinT - Best Known (88−53, 88, s)-Nets in Base 16

Best Known (88−53, 88, s)-Nets in Base 16

(88−53, 88, 103)-Net over F₁₆ — Constructive and digital

Digital (35, 88, 103)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (3, 29, 38)-net over F₁₆, using
  - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]
2. digital (6, 59, 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]

(88−53, 88, 128)-Net in Base 16 — Constructive

(35, 88, 128)-net in base 16, using

2 times m-reduction [i] based on (35, 90, 128)-net in base 16, using
- base change [i] based on digital (5, 60, 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]

(88−53, 88, 193)-Net over F₁₆ — Digital

Digital (35, 88, 193)-net over F₁₆, using

t-expansion [i] based on digital (33, 88, 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]

(88−53, 88, 7508)-Net in Base 16 — Upper bound on s

There is no (35, 88, 7509)-net in base 16, because

1 times m-reduction [i] would yield (35, 87, 7509)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 574 462013 726791 051108 134688 175236 060874 196132 037211 521406 610986 850200 414086 573947 283583 362666 863369 969136 > 16⁸⁷ [i]