MinT - Best Known (113−55, 113, s)-Nets in Base 16

Best Known (113−55, 113, s)-Nets in Base 16

(113−55, 113, 516)-Net over F₁₆ — Constructive and digital

Digital (58, 113, 516)-net over F₁₆, using

1 times m-reduction [i] based on digital (58, 114, 516)-net over F₁₆, using
- trace code for nets [i] based on digital (1, 57, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(113−55, 113, 578)-Net over F₁₆ — Digital

Digital (58, 113, 578)-net over F₁₆, using

1 times m-reduction [i] based on digital (58, 114, 578)-net over F₁₆, using
- trace code for nets [i] based on digital (1, 57, 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]

(113−55, 113, 71960)-Net in Base 16 — Upper bound on s

There is no (58, 113, 71961)-net in base 16, because

1 times m-reduction [i] would yield (58, 112, 71961)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 726 970180 778745 191679 044113 269175 116320 182423 514421 621751 033439 963100 935209 804511 098417 919300 729935 274063 795949 323286 920150 834355 149056 > 16¹¹² [i]