MinT - Best Known (32, 32+87, s)-Nets in Base 16

Best Known (32, 32+87, s)-Nets in Base 16

(32, 32+87, 65)-Net over F₁₆ — Constructive and digital

Digital (32, 119, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 119, 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+87, 98)-Net in Base 16 — Constructive

(32, 119, 98)-net in base 16, using

6 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+87, 168)-Net over F₁₆ — Digital

Digital (32, 119, 168)-net over F₁₆, using

t-expansion [i] based on digital (31, 119, 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+87, 2244)-Net in Base 16 — Upper bound on s

There is no (32, 119, 2245)-net in base 16, because

1 times m-reduction [i] would yield (32, 118, 2245)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 12304 179280 149235 953432 750857 003793 465125 099314 478565 891229 949456 076486 709408 251448 655718 886272 449380 865760 837609 321717 238664 120750 694746 639776 > 16¹¹⁸ [i]