MinT - Best Known (106−57, 106, s)-Nets in Base 2

Best Known (106−57, 106, s)-Nets in Base 2

(106−57, 106, 35)-Net over F₂ — Constructive and digital

Digital (49, 106, 35)-net over F₂, using

t-expansion [i] based on digital (48, 106, 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]

(106−57, 106, 36)-Net over F₂ — Digital

Digital (49, 106, 36)-net over F₂, using

t-expansion [i] based on digital (47, 106, 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]

(106−57, 106, 107)-Net over F₂ — Upper bound on s (digital)

There is no digital (49, 106, 108)-net over F₂, because

7 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]

(106−57, 106, 108)-Net in Base 2 — Upper bound on s

There is no (49, 106, 109)-net in base 2, because

3 times m-reduction [i] would yield (49, 103, 109)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2¹⁰³, 109, S₂, 54), but
  - adding a parity check bit [i] would yield OA(2¹⁰⁴, 110, S₂, 55), but
    - the (dual) Plotkin bound shows that MÂ â‰¥ 162 259276 829213 363391 578010 288128 / 7 > 2¹⁰⁴ [i]