MinT - Best Known (115−87, 115, s)-Nets in Base 16

Best Known (115−87, 115, s)-Nets in Base 16

(115−87, 115, 65)-Net over F₁₆ — Constructive and digital

Digital (28, 115, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 115, 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]

(115−87, 115, 76)-Net in Base 16 — Constructive

(28, 115, 76)-net in base 16, using

base change [i] based on digital (5, 92, 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]

(115−87, 115, 156)-Net over F₁₆ — Digital

Digital (28, 115, 156)-net over F₁₆, using

t-expansion [i] based on digital (27, 115, 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]

(115−87, 115, 1728)-Net in Base 16 — Upper bound on s

There is no (28, 115, 1729)-net in base 16, because

1 times m-reduction [i] would yield (28, 114, 1729)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 186691 470194 307899 668578 752357 670369 549285 184669 013121 104833 505640 078465 657197 292996 787182 814421 710774 538180 893289 399588 108887 736315 899456 > 16¹¹⁴ [i]