MinT - Best Known (64−62, 64, s)-Nets in Base 32

Best Known (64−62, 64, s)-Nets in Base 32

(64−62, 64, 44)-Net over F₃₂ — Constructive and digital

Digital (2, 64, 44)-net over F₃₂, using

t-expansion [i] based on digital (1, 64, 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]

(64−62, 64, 53)-Net over F₃₂ — Digital

Digital (2, 64, 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]

(64−62, 64, 129)-Net over F₃₂ — Upper bound on s (digital)

There is no digital (2, 64, 130)-net over F₃₂, because

extracting embedded orthogonal array [i] would yield linear OA(32⁶⁴, 130, F₃₂, 62) (dual of [130, 66, 63]-code), but
- constructionÂ Y1 [i] would yield
  1. linear OA(32⁶³, 66, F₃₂, 62) (dual of [66, 3, 63]-code), but
    - bound for almost-MDS-codes with dimension nÂ =Â 3 [i]
  2. linear OA(32⁶⁶, 130, F₃₂, 64) (dual of [130, 64, 65]-code), but
    - discarding factors / shortening the dual code 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]

(64−62, 64, 246)-Net in Base 32 — Upper bound on s

There is no (2, 64, 247)-net in base 32, because

3 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]