MinT - Best Known (97−72, 97, s)-Nets in Base 16

Best Known (97−72, 97, s)-Nets in Base 16

(97−72, 97, 65)-Net over F₁₆ — Constructive and digital

Digital (25, 97, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 97, 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]

(97−72, 97, 76)-Net in Base 16 — Constructive

(25, 97, 76)-net in base 16, using

3 times m-reduction [i] based on (25, 100, 76)-net in base 16, using
- base change [i] based on digital (5, 80, 76)-net over F₃₂, using
  - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]

(97−72, 97, 144)-Net over F₁₆ — Digital

Digital (25, 97, 144)-net over F₁₆, using

net from sequence [i] based on digital (25, 143)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 25 and N(F) ≥ 144, using
  - a curve from the manYPoints database [i]

(97−72, 97, 1651)-Net in Base 16 — Upper bound on s

There is no (25, 97, 1652)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 636 458938 813961 008887 379780 358871 358661 815111 861715 782295 060303 433667 253418 904415 599282 728941 388315 595591 762913 147956 > 16⁹⁷ [i]