MinT - Best Known (76−40, 76, s)-Nets in Base 16

Best Known (76−40, 76, s)-Nets in Base 16

(76−40, 76, 130)-Net over F₁₆ — Constructive and digital

Digital (36, 76, 130)-net over F₁₆, using

8 times m-reduction [i] based on digital (36, 84, 130)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 30, 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, 54, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆ (see above)

(76−40, 76, 192)-Net in Base 16 — Constructive

(36, 76, 192)-net in base 16, using

1 times m-reduction [i] based on (36, 77, 192)-net in base 16, using
- base change [i] based on digital (3, 44, 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]

(76−40, 76, 223)-Net over F₁₆ — Digital

Digital (36, 76, 223)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(76−40, 76, 225)-Net in Base 16

(36, 76, 225)-net in base 16, using

2 times m-reduction [i] based on (36, 78, 225)-net in base 16, using
- base change [i] based on digital (10, 52, 225)-net over F₆₄, using
  - net from sequence [i] based on digital (10, 224)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 10 and N(F) ≥ 225, using
      - a curve from the manYPoints database [i]

(76−40, 76, 20827)-Net in Base 16 — Upper bound on s

There is no (36, 76, 20828)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 32 596275 988980 135017 201215 973705 610109 335953 066141 782879 235207 751560 592352 215706 783604 145526 > 16⁷⁶ [i]