MinT - Best Known (64−35, 64, s)-Nets in Base 16

Best Known (64−35, 64, s)-Nets in Base 16

(64−35, 64, 130)-Net over F₁₆ — Constructive and digital

Digital (29, 64, 130)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 23, 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, 41, 65)-net over F₁₆, using
  - net from sequence [i] based on digital (6, 64)-sequence over F₁₆ (see above)

(64−35, 64, 162)-Net over F₁₆ — Digital

Digital (29, 64, 162)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(64−35, 64, 177)-Net in Base 16 — Constructive

(29, 64, 177)-net in base 16, using

2 times m-reduction [i] based on (29, 66, 177)-net in base 16, using
- base change [i] based on digital (7, 44, 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]

(64−35, 64, 13864)-Net in Base 16 — Upper bound on s

There is no (29, 64, 13865)-net in base 16, because

1 times m-reduction [i] would yield (29, 63, 13865)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 7244 055068 376930 295788 616499 268370 351543 730691 381047 836716 066356 542534 462576 > 16⁶³ [i]