MinT - Best Known (101−52, 101, s)-Nets in Base 2

Best Known (101−52, 101, s)-Nets in Base 2

(101−52, 101, 35)-Net over F₂ — Constructive and digital

Digital (49, 101, 35)-net over F₂, using

t-expansion [i] based on digital (48, 101, 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]

(101−52, 101, 36)-Net over F₂ — Digital

Digital (49, 101, 36)-net over F₂, using

t-expansion [i] based on digital (47, 101, 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]

(101−52, 101, 107)-Net over F₂ — Upper bound on s (digital)

There is no digital (49, 101, 108)-net over F₂, because

2 times m-reduction [i] would yield digital (49, 99, 108)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁹⁹, 108, F₂, 50) (dual of [108, 9, 51]-code), but
  - residual code [i] would yield linear OA(2⁴⁹, 57, F₂, 25) (dual of [57, 8, 26]-code), but
    - â€œBJVâ€ bound on codes from Brouwerâ€™s database [i]

(101−52, 101, 109)-Net in Base 2 — Upper bound on s

There is no (49, 101, 110)-net in base 2, because

2 times m-reduction [i] would yield (49, 99, 110)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁹⁹, 110, S₂, 50), but
  - the linear programming bound shows that MÂ â‰¥ 318 814125 957399 694476 420856 152064 / 403 > 2⁹⁹ [i]