MinT - Best Known (33−32, 33, s)-Nets in Base 32

Best Known (33−32, 33, s)-Nets in Base 32

(33−32, 33, 44)-Net over F₃₂ — Constructive and digital

Digital (1, 33, 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]

(33−32, 33, 95)-Net over F₃₂ — Upper bound on s (digital)

There is no digital (1, 33, 96)-net over F₃₂, because

extracting embedded orthogonal array [i] would yield linear OA(32³³, 96, F₃₂, 32) (dual of [96, 63, 33]-code), but
- dual of a near-MDS code is again a near-MDS code [i] would yield linear OA(32⁶³, 96, F₃₂, 62) (dual of [96, 33, 63]-code), but
  - discarding factors / shortening the dual code would yield linear OA(32⁶³, 66, F₃₂, 62) (dual of [66, 3, 63]-code), but
    - bound for almost-MDS-codes with dimension nÂ =Â 3 [i]

(33−32, 33, 165)-Net in Base 32 — Upper bound on s

There is no (1, 33, 166)-net in base 32, because

24 times m-reduction [i] would yield (1, 9, 166)-net in base 32, but
- extracting embedded orthogonal array [i] would yield OA(32⁹, 166, S₃₂, 8), but
  - the linear programming bound shows that MÂ â‰¥ 2 187320 464516 698429 456384 / 61939 225499 > 32⁹ [i]