MinT - Best Known (105−51, 105, s)-Nets in Base 16

Best Known (105−51, 105, s)-Nets in Base 16

(105−51, 105, 516)-Net over F₁₆ — Constructive and digital

Digital (54, 105, 516)-net over F₁₆, using

1 times m-reduction [i] based on digital (54, 106, 516)-net over F₁₆, using
- trace code for nets [i] based on digital (1, 53, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(105−51, 105, 578)-Net over F₁₆ — Digital

Digital (54, 105, 578)-net over F₁₆, using

1 times m-reduction [i] based on digital (54, 106, 578)-net over F₁₆, using
- trace code for nets [i] based on digital (1, 53, 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]

(105−51, 105, 69276)-Net in Base 16 — Upper bound on s

There is no (54, 105, 69277)-net in base 16, because

1 times m-reduction [i] would yield (54, 104, 69277)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 169240 455819 968562 754643 841984 108102 006054 688334 058640 810831 160707 322009 911622 629132 678311 929827 683954 706310 318817 521989 543876 > 16¹⁰⁴ [i]