MinT - Best Known (112−55, 112, s)-Nets in Base 16

Best Known (112−55, 112, s)-Nets in Base 16

(112−55, 112, 516)-Net over F₁₆ — Constructive and digital

Digital (57, 112, 516)-net over F₁₆, using

trace code for nets [i] based on digital (1, 56, 258)-net over F₂₅₆, using
- net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(112−55, 112, 578)-Net over F₁₆ — Digital

Digital (57, 112, 578)-net over F₁₆, using

trace code for nets [i] based on digital (1, 56, 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]

(112−55, 112, 64936)-Net in Base 16 — Upper bound on s

There is no (57, 112, 64937)-net in base 16, because

1 times m-reduction [i] would yield (57, 111, 64937)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 45 440511 360429 037485 336108 658302 356798 721913 415639 453297 852751 487227 161266 593642 023413 195590 451417 062949 980028 982572 235269 356969 509536 > 16¹¹¹ [i]