MinT - Best Known (53−51, 53, s)-Nets in Base 32

Best Known (53−51, 53, s)-Nets in Base 32

(53−51, 53, 44)-Net over F₃₂ — Constructive and digital

Digital (2, 53, 44)-net over F₃₂, using

t-expansion [i] based on digital (1, 53, 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]

(53−51, 53, 53)-Net over F₃₂ — Digital

Digital (2, 53, 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]

(53−51, 53, 248)-Net in Base 32 — Upper bound on s

There is no (2, 53, 249)-net in base 32, because

extracting embedded orthogonal array [i] would yield OA(32⁵³, 249, S₃₂, 51), but
- the linear programming bound shows that MÂ â‰¥ 1 415329 045691 058293 440323 562599 182789 070899 927587 626006 470680 651402 143742 922493 081536 303831 449600 / 23691 100188 925591 > 32⁵³ [i]