MinT - Best Known (32, 32+89, s)-Nets in Base 16

Best Known (32, 32+89, s)-Nets in Base 16

(32, 32+89, 65)-Net over F₁₆ — Constructive and digital

Digital (32, 121, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 121, 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]

(32, 32+89, 98)-Net in Base 16 — Constructive

(32, 121, 98)-net in base 16, using

4 times m-reduction [i] based on (32, 125, 98)-net in base 16, using
- base change [i] based on digital (7, 100, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
      - a function field by SÃ©mirat [i]

(32, 32+89, 168)-Net over F₁₆ — Digital

Digital (32, 121, 168)-net over F₁₆, using

t-expansion [i] based on digital (31, 121, 168)-net over F₁₆, using
- net from sequence [i] based on digital (31, 167)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 31 and N(F) ≥ 168, using
    - a curve from the manYPoints database [i]

(32, 32+89, 2187)-Net in Base 16 — Upper bound on s

There is no (32, 121, 2188)-net in base 16, because

1 times m-reduction [i] would yield (32, 120, 2188)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 3 141441 793748 518335 308939 810231 184103 404681 024531 540608 955897 485219 220405 056281 481366 799017 383720 169227 225213 328720 828541 423495 226208 286018 089056 > 16¹²⁰ [i]