MinT - Best Known (78−46, 78, s)-Nets in Base 16

Best Known (78−46, 78, s)-Nets in Base 16

(78−46, 78, 103)-Net over F₁₆ — Constructive and digital

Digital (32, 78, 103)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (3, 26, 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, 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]

(78−46, 78, 128)-Net in Base 16 — Constructive

(32, 78, 128)-net in base 16, using

3 times m-reduction [i] based on (32, 81, 128)-net in base 16, using
- base change [i] based on digital (5, 54, 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]

(78−46, 78, 168)-Net over F₁₆ — Digital

Digital (32, 78, 168)-net over F₁₆, using

t-expansion [i] based on digital (31, 78, 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]

(78−46, 78, 7606)-Net in Base 16 — Upper bound on s

There is no (32, 78, 7607)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 8355 072807 332627 774807 047171 872323 006369 189926 052191 937358 672324 508478 467851 996584 370297 496816 > 16⁷⁸ [i]