MinT - Best Known (31, 31+97, s)-Nets in Base 16

Best Known (31, 31+97, s)-Nets in Base 16

(31, 31+97, 65)-Net over F₁₆ — Constructive and digital

Digital (31, 128, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 128, 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]

(31, 31+97, 76)-Net in Base 16 — Constructive

(31, 128, 76)-net in base 16, using

2 times m-reduction [i] based on (31, 130, 76)-net in base 16, using
- base change [i] based on digital (5, 104, 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]

(31, 31+97, 168)-Net over F₁₆ — Digital

Digital (31, 128, 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]

(31, 31+97, 1890)-Net in Base 16 — Upper bound on s

There is no (31, 128, 1891)-net in base 16, because

1 times m-reduction [i] would yield (31, 127, 1891)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 853 000138 230299 669743 972510 721718 248270 332998 900190 437948 617230 444251 739811 716524 247051 509355 839776 833842 714041 797520 447369 456075 369067 725975 842549 809196 > 16¹²⁷ [i]