MinT - Best Known (66−34, 66, s)-Nets in Base 16

Best Known (66−34, 66, s)-Nets in Base 16

(66−34, 66, 130)-Net over F₁₆ — Constructive and digital

Digital (32, 66, 130)-net over F₁₆, using

6 times m-reduction [i] based on digital (32, 72, 130)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 26, 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, 46, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆ (see above)

(66−34, 66, 192)-Net in Base 16 — Constructive

(32, 66, 192)-net in base 16, using

base change [i] based on (10, 44, 192)-net in base 64, using
- 5 times m-reduction [i] based on (10, 49, 192)-net in base 64, using
  - base change [i] based on digital (3, 42, 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]

(66−34, 66, 225)-Net over F₁₆ — Digital

Digital (32, 66, 225)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(66−34, 66, 22620)-Net in Base 16 — Upper bound on s

There is no (32, 66, 22621)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 29 659477 320081 346518 210759 506813 789127 142071 807006 473265 458774 359781 366323 496956 > 16⁶⁶ [i]