MinT - Best Known (68−67, 68, s)-Nets in Base 32

Best Known (68−67, 68, s)-Nets in Base 32

(68−67, 68, 44)-Net over F₃₂ — Constructive and digital

Digital (1, 68, 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]

(68−67, 68, 65)-Net over F₃₂ — Upper bound on s (digital)

There is no digital (1, 68, 66)-net over F₃₂, because

5 times m-reduction [i] would yield digital (1, 63, 66)-net over F₃₂, but
- extracting embedded orthogonal array [i] 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]

(68−67, 68, 66)-Net in Base 32 — Upper bound on s

There is no (1, 68, 67)-net in base 32, because

3 times m-reduction [i] would yield (1, 65, 67)-net in base 32, but
- extracting embedded orthogonal array [i] would yield OA(32⁶⁵, 67, S₃₂, 64), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 6561 752174 349035 773117 506681 352864 096059 508292 679637 309277 311819 229858 997598 127769 670539 531069 161472 / 65 > 32⁶⁵ [i]