MinT - Best Known (82−40, 82, s)-Nets in Base 16

Best Known (82−40, 82, s)-Nets in Base 16

(82−40, 82, 516)-Net over F₁₆ — Constructive and digital

Digital (42, 82, 516)-net over F₁₆, using

trace code for nets [i] based on digital (1, 41, 258)-net over F₂₅₆, using
- net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(82−40, 82, 578)-Net over F₁₆ — Digital

Digital (42, 82, 578)-net over F₁₆, using

trace code for nets [i] based on digital (1, 41, 289)-net over F₂₅₆, using
- net from sequence [i] based on digital (1, 288)-sequence over F₂₅₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289, using
    - sharp bound on the number of rational points on curves with genus 1 [i]

(82−40, 82, 47863)-Net in Base 16 — Upper bound on s

There is no (42, 82, 47864)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 546 821965 308390 663952 954494 436702 424009 086152 050831 768937 127990 293765 349833 417845 052695 806701 454201 > 16⁸² [i]