MinT - Best Known (79−46, 79, s)-Nets in Base 16

Best Known (79−46, 79, s)-Nets in Base 16

(79−46, 79, 110)-Net over F₁₆ — Constructive and digital

Digital (33, 79, 110)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (4, 27, 45)-net over F₁₆, using
  - net from sequence [i] based on digital (4, 44)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 4 and N(F) ≥ 45, using
      - a function field by GarcÃa and Quoos [i]
2. digital (6, 52, 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]

(79−46, 79, 128)-Net in Base 16 — Constructive

(33, 79, 128)-net in base 16, using

5 times m-reduction [i] based on (33, 84, 128)-net in base 16, using
- base change [i] based on digital (5, 56, 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]

(79−46, 79, 193)-Net over F₁₆ — Digital

Digital (33, 79, 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]

(79−46, 79, 8582)-Net in Base 16 — Upper bound on s

There is no (33, 79, 8583)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 133620 837310 394984 913840 870007 414888 004921 260234 023040 021599 130052 996432 258057 221762 961681 287936 > 16⁷⁹ [i]