MinT - Best Known (31, 31+37, s)-Nets in Base 16

Best Known (31, 31+37, s)-Nets in Base 16

(31, 31+37, 130)-Net over F₁₆ — Constructive and digital

Digital (31, 68, 130)-net over F₁₆, using

1 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+37, 174)-Net over F₁₆ — Digital

Digital (31, 68, 174)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(31, 31+37, 177)-Net in Base 16 — Constructive

(31, 68, 177)-net in base 16, using

4 times m-reduction [i] based on (31, 72, 177)-net in base 16, using
- base change [i] based on digital (7, 48, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(31, 31+37, 15267)-Net in Base 16 — Upper bound on s

There is no (31, 68, 15268)-net in base 16, because

1 times m-reduction [i] would yield (31, 67, 15268)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 474 618243 321034 662678 399805 311783 347740 317916 138744 395725 821331 161040 480857 537786 > 16⁶⁷ [i]