MinT - Best Known (37, 78, s)-Nets in Base 16

Best Known (37, 78, s)-Nets in Base 16

(37, 78, 130)-Net over F₁₆ — Constructive and digital

Digital (37, 78, 130)-net over F₁₆, using

9 times m-reduction [i] based on digital (37, 87, 130)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 31, 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]
  2. digital (6, 56, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆ (see above)

(37, 78, 192)-Net in Base 16 — Constructive

(37, 78, 192)-net in base 16, using

16¹ times duplication [i] based on (36, 77, 192)-net in base 16, using
- base change [i] based on digital (3, 44, 192)-net over F₁₂₈, using
  - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
      - a function field by SÃ©mirat [i]

(37, 78, 229)-Net over F₁₆ — Digital

Digital (37, 78, 229)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(37, 78, 23926)-Net in Base 16 — Upper bound on s

There is no (37, 78, 23927)-net in base 16, because

1 times m-reduction [i] would yield (37, 77, 23927)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 521 671228 863286 991812 365931 092496 120244 041103 076761 602305 605334 910796 697812 158675 587171 890101 > 16⁷⁷ [i]