MinT - Best Known (67−65, 67, s)-Nets in Base 32

Best Known (67−65, 67, s)-Nets in Base 32

(67−65, 67, 44)-Net over F₃₂ — Constructive and digital

Digital (2, 67, 44)-net over F₃₂, using

t-expansion [i] based on digital (1, 67, 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]

(67−65, 67, 53)-Net over F₃₂ — Digital

Digital (2, 67, 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]

(67−65, 67, 98)-Net over F₃₂ — Upper bound on s (digital)

There is no digital (2, 67, 99)-net over F₃₂, because

1 times m-reduction [i] would yield digital (2, 66, 99)-net over F₃₂, but
- extracting embedded orthogonal array [i] would yield linear OA(32⁶⁶, 99, F₃₂, 64) (dual of [99, 33, 65]-code), but
  - residual code [i] would yield OA(32², 34, S₃₂, 2), but
    - bound for OAs with strength kÂ =Â 2 [i]
    - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 1055 > 32² [i]

(67−65, 67, 246)-Net in Base 32 — Upper bound on s

There is no (2, 67, 247)-net in base 32, because

6 times m-reduction [i] would yield (2, 61, 247)-net in base 32, but
- extracting embedded orthogonal array [i] would yield OA(32⁶¹, 247, S₃₂, 59), but
  - the linear programming bound shows that MÂ â‰¥ 69666 313688 466493 132585 748412 717479 501180 313532 643690 366794 004047 936261 038394 666402 878176 578856 512986 808320 / 1067 161918 947067 > 32⁶¹ [i]