MinT - Best Known (177−128, 177, s)-Nets in Base 2

Best Known (177−128, 177, s)-Nets in Base 2

(177−128, 177, 35)-Net over F₂ — Constructive and digital

Digital (49, 177, 35)-net over F₂, using

t-expansion [i] based on digital (48, 177, 35)-net over F₂, using
- net from sequence [i] based on digital (48, 34)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 41, N(F)Â =Â 32, 1 place with degree 2, and 2 places with degree 4 [i] based on function field F/F₂ with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]

(177−128, 177, 36)-Net over F₂ — Digital

Digital (49, 177, 36)-net over F₂, using

t-expansion [i] based on digital (47, 177, 36)-net over F₂, using
- net from sequence [i] based on digital (47, 35)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 47 and N(F) ≥ 36, using
    - Drinfeld modules of rank 1 [i]

(177−128, 177, 73)-Net in Base 2 — Upper bound on s

There is no (49, 177, 74)-net in base 2, because

35 times m-reduction [i] would yield (49, 142, 74)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2¹⁴², 74, S₂, 2, 93), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 267 608942 382367 477698 428619 271780 338071 764992 / 47 > 2¹⁴² [i]