MinT - Best Known (68−35, 68, s)-Nets in Base 16

Best Known (68−35, 68, s)-Nets in Base 16

(68−35, 68, 130)-Net over F₁₆ — Constructive and digital

Digital (33, 68, 130)-net over F₁₆, using

7 times m-reduction [i] based on digital (33, 75, 130)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 27, 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, 48, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆ (see above)

(68−35, 68, 192)-Net in Base 16 — Constructive

(33, 68, 192)-net in base 16, using

2 times m-reduction [i] based on (33, 70, 192)-net in base 16, using
- base change [i] based on digital (3, 40, 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]

(68−35, 68, 231)-Net over F₁₆ — Digital

Digital (33, 68, 231)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(68−35, 68, 26628)-Net in Base 16 — Upper bound on s

There is no (33, 68, 26629)-net in base 16, because

1 times m-reduction [i] would yield (33, 67, 26629)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 474 293383 201795 138268 924020 605190 073005 945629 151844 543783 619730 216562 989192 208896 > 16⁶⁷ [i]