MinT - Best Known (67, 67+63, s)-Nets in Base 16

Best Known (67, 67+63, s)-Nets in Base 16

(67, 67+63, 518)-Net over F₁₆ — Constructive and digital

Digital (67, 130, 518)-net over F₁₆, using

trace code for nets [i] based on digital (2, 65, 259)-net over F₂₅₆, using
- net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(67, 67+63, 642)-Net over F₁₆ — Digital

Digital (67, 130, 642)-net over F₁₆, using

trace code for nets [i] based on digital (2, 65, 321)-net over F₂₅₆, using
- net from sequence [i] based on digital (2, 320)-sequence over F₂₅₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321, using
    - sharp bound on the number of rational points on curves with genus 2 [i]

(67, 67+63, 84829)-Net in Base 16 — Upper bound on s

There is no (67, 130, 84830)-net in base 16, because

1 times m-reduction [i] would yield (67, 129, 84830)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 214560 084852 948181 535409 072988 948438 833636 927175 424086 745413 188065 465457 349386 272585 972877 322758 290745 546333 230157 277305 400614 066889 794265 067229 242152 396576 > 16¹²⁹ [i]