MinT - Best Known (27, 27+89, s)-Nets in Base 16

Best Known (27, 27+89, s)-Nets in Base 16

(27, 27+89, 65)-Net over F₁₆ — Constructive and digital

Digital (27, 116, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 116, 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]

(27, 27+89, 66)-Net in Base 16 — Constructive

(27, 116, 66)-net in base 16, using

t-expansion [i] based on (25, 116, 66)-net in base 16, using
- net from sequence [i] based on (25, 65)-sequence in base 16, using
  - base expansion [i] based on digital (50, 65)-sequence over F₄, using
    - t-expansion [i] based on digital (49, 65)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
        T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(27, 27+89, 156)-Net over F₁₆ — Digital

Digital (27, 116, 156)-net over F₁₆, using

net from sequence [i] based on digital (27, 155)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 27 and N(F) ≥ 156, using
  - a curve from the manYPoints database [i]

(27, 27+89, 1589)-Net in Base 16 — Upper bound on s

There is no (27, 116, 1590)-net in base 16, because

1 times m-reduction [i] would yield (27, 115, 1590)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 986905 843032 879326 791084 935427 920840 364365 573795 783006 524996 095730 227747 860578 838280 586407 018210 409620 274593 598572 347375 433813 733295 184526 > 16¹¹⁵ [i]