MinT - Best Known (57−55, 57, s)-Nets in Base 32

Best Known (57−55, 57, s)-Nets in Base 32

(57−55, 57, 44)-Net over F₃₂ — Constructive and digital

Digital (2, 57, 44)-net over F₃₂, using

t-expansion [i] based on digital (1, 57, 44)-net over F₃₂, using
- net from sequence [i] based on digital (1, 43)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 1 and N(F) ≥ 44, using
    - a function field by SÃ©mirat [i]

(57−55, 57, 53)-Net over F₃₂ — Digital

Digital (2, 57, 53)-net over F₃₂, using

net from sequence [i] based on digital (2, 52)-sequence over F₃₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 2 and N(F) ≥ 53, using
  - sharp bound on the number of rational points on curves with genus 2 [i]

(57−55, 57, 247)-Net in Base 32 — Upper bound on s

There is no (2, 57, 248)-net in base 32, because

2 times m-reduction [i] would yield (2, 55, 248)-net in base 32, but
- extracting embedded orthogonal array [i] would yield OA(32⁵⁵, 248, S₃₂, 53), but
  - the linear programming bound shows that MÂ â‰¥ 16062 875815 361622 591832 993088 417824 992293 220291 564835 849555 692861 231558 994841 916425 250457 423034 762815 078400 / 262402 337018 559377 249213 > 32⁵⁵ [i]