MinT - Best Known (177−145, 177, s)-Nets in Base 2

Best Known (177−145, 177, s)-Nets in Base 2

(177−145, 177, 21)-Net over F₂ — Constructive and digital

Digital (32, 177, 21)-net over F₂, using

t-expansion [i] based on digital (21, 177, 21)-net over F₂, using
- net from sequence [i] based on digital (21, 20)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 21 and N(F) ≥ 21, using
    - an explicitly constructive algebraic function field [i]

(177−145, 177, 27)-Net over F₂ — Digital

Digital (32, 177, 27)-net over F₂, using

t-expansion [i] based on digital (31, 177, 27)-net over F₂, using
- net from sequence [i] based on digital (31, 26)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 31 and N(F) ≥ 27, using
    - Drinfeld modules of rank 1 [i]

(177−145, 177, 41)-Net in Base 2 — Upper bound on s

There is no (32, 177, 42)-net in base 2, because

16 times m-reduction [i] would yield (32, 161, 42)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2¹⁶¹, 42, S₂, 4, 129), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 257 224288 170238 913603 848530 558065 811459 444127 563776 / 65 > 2¹⁶¹ [i]