MinT - Best Known (113−93, 113, s)-Nets in Base 16

Best Known (113−93, 113, s)-Nets in Base 16

(113−93, 113, 65)-Net over F₁₆ — Constructive and digital

Digital (20, 113, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 113, 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]

(113−93, 113, 129)-Net over F₁₆ — Digital

Digital (20, 113, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 113, 129)-net over F₁₆, using
- net from sequence [i] based on digital (19, 128)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 19 and N(F) ≥ 129, using
    - a curve from the manYPoints database [i]

(113−93, 113, 1000)-Net in Base 16 — Upper bound on s

There is no (20, 113, 1001)-net in base 16, because

1 times m-reduction [i] would yield (20, 112, 1001)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 759 264525 385508 730352 858113 275760 300466 404890 637673 552631 506587 234075 738623 096388 394905 438940 166746 623410 347296 421326 052003 424988 574816 > 16¹¹² [i]