MinT - Best Known (62−33, 62, s)-Nets in Base 16

Best Known (62−33, 62, s)-Nets in Base 16

(62−33, 62, 130)-Net over F₁₆ — Constructive and digital

Digital (29, 62, 130)-net over F₁₆, using

1 times m-reduction [i] based on digital (29, 63, 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, 40, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆ (see above)

(62−33, 62, 177)-Net in Base 16 — Constructive

(29, 62, 177)-net in base 16, using

4 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]

(62−33, 62, 182)-Net over F₁₆ — Digital

Digital (29, 62, 182)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(62−33, 62, 17658)-Net in Base 16 — Upper bound on s

There is no (29, 62, 17659)-net in base 16, because

1 times m-reduction [i] would yield (29, 61, 17659)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 28 292279 819123 973944 676842 605877 722198 903671 686049 779724 625498 037318 115661 > 16⁶¹ [i]