MinT - Best Known (55, 55+51, s)-Nets in Base 16

Best Known (55, 55+51, s)-Nets in Base 16

(55, 55+51, 518)-Net over F₁₆ — Constructive and digital

Digital (55, 106, 518)-net over F₁₆, using

trace code for nets [i] based on digital (2, 53, 259)-net over F₂₅₆, using
- net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(55, 55+51, 642)-Net over F₁₆ — Digital

Digital (55, 106, 642)-net over F₁₆, using

trace code for nets [i] based on digital (2, 53, 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]

(55, 55+51, 77403)-Net in Base 16 — Upper bound on s

There is no (55, 106, 77404)-net in base 16, because

1 times m-reduction [i] would yield (55, 105, 77404)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 707931 382429 087324 394150 247703 762935 469406 319878 707254 752071 381249 451984 354388 942472 864631 202660 816106 037938 540931 557046 576501 > 16¹⁰⁵ [i]