MinT - Best Known (31, 31+33, s)-Nets in Base 16

Best Known (31, 31+33, s)-Nets in Base 16

(31, 31+33, 130)-Net over F₁₆ — Constructive and digital

Digital (31, 64, 130)-net over F₁₆, using

5 times m-reduction [i] based on digital (31, 69, 130)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 25, 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, 44, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆ (see above)

(31, 31+33, 192)-Net in Base 16 — Constructive

(31, 64, 192)-net in base 16, using

16¹ times duplication [i] based on (30, 63, 192)-net in base 16, using
- base change [i] based on digital (3, 36, 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]

(31, 31+33, 219)-Net over F₁₆ — Digital

Digital (31, 64, 219)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(31, 31+33, 24975)-Net in Base 16 — Upper bound on s

There is no (31, 64, 24976)-net in base 16, because

1 times m-reduction [i] would yield (31, 63, 24976)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 7237 515063 440020 587122 804355 062367 973070 165212 498879 832985 891004 042933 470616 > 16⁶³ [i]