MinT - Best Known (125−93, 125, s)-Nets in Base 16

Best Known (125−93, 125, s)-Nets in Base 16

(125−93, 125, 65)-Net over F₁₆ — Constructive and digital

Digital (32, 125, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 125, 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]

(125−93, 125, 98)-Net in Base 16 — Constructive

(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]

(125−93, 125, 168)-Net over F₁₆ — Digital

Digital (32, 125, 168)-net over F₁₆, using

t-expansion [i] based on digital (31, 125, 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]

(125−93, 125, 2088)-Net in Base 16 — Upper bound on s

There is no (32, 125, 2089)-net in base 16, because

1 times m-reduction [i] would yield (32, 124, 2089)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 208493 127094 088145 958825 039250 898753 171389 674911 506637 328536 357049 589977 541748 365581 040307 934355 498747 637027 333881 504377 439672 528567 493566 072064 449536 > 16¹²⁴ [i]