MinT - Best Known (20, 20+53, s)-Nets in Base 16

Best Known (20, 20+53, s)-Nets in Base 16

(20, 20+53, 65)-Net over F₁₆ — Constructive and digital

Digital (20, 73, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 73, 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]

(20, 20+53, 76)-Net in Base 16 — Constructive

(20, 73, 76)-net in base 16, using

2 times m-reduction [i] based on (20, 75, 76)-net in base 16, using
- base change [i] based on digital (5, 60, 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]

(20, 20+53, 129)-Net over F₁₆ — Digital

Digital (20, 73, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 73, 129)-net over F₁₆, using
- net from sequence [i] based on digital (19, 128)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 19 and N(F) ≥ 129, using
    - a curve from the manYPoints database [i]

(20, 20+53, 1505)-Net in Base 16 — Upper bound on s

There is no (20, 73, 1506)-net in base 16, because

1 times m-reduction [i] would yield (20, 72, 1506)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 503 698386 851382 679906 747130 370519 045008 642630 898314 618834 918585 477005 652529 668282 179216 > 16⁷² [i]