MinT - Best Known (78, 163, s)-Nets in Base 2

Best Known (78, 163, s)-Nets in Base 2

(78, 163, 50)-Net over F₂ — Constructive and digital

Digital (78, 163, 50)-net over F₂, using

t-expansion [i] based on digital (75, 163, 50)-net over F₂, using
- net from sequence [i] based on digital (75, 49)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 69, N(F)Â =Â 48, 1 place with degree 2, and 1 place with degree 6 [i] based on function field F/F₂ with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]

(78, 163, 52)-Net over F₂ — Digital

Digital (78, 163, 52)-net over F₂, using

t-expansion [i] based on digital (77, 163, 52)-net over F₂, using
- net from sequence [i] based on digital (77, 51)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 77 and N(F) ≥ 52, using
    - Drinfeld modules of rank 1 [i]

(78, 163, 165)-Net over F₂ — Upper bound on s (digital)

There is no digital (78, 163, 166)-net over F₂, because

5 times m-reduction [i] would yield digital (78, 158, 166)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁵⁸, 166, F₂, 80) (dual of [166, 8, 81]-code), but
  - residual code [i] would yield linear OA(2⁷⁸, 85, F₂, 40) (dual of [85, 7, 41]-code), but
    - â€œHelâ€ bound on codes from Brouwerâ€™s database [i]

(78, 163, 167)-Net in Base 2 — Upper bound on s

There is no (78, 163, 168)-net in base 2, because

1 times m-reduction [i] would yield (78, 162, 168)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2¹⁶², 168, S₂, 84), but
  - adding a parity check bit [i] would yield OA(2¹⁶³, 169, S₂, 85), but
    - the (dual) Plotkin bound shows that MÂ â‰¥ 561 216628 735066 720590 214975 763052 679547 878096 502784 / 43 > 2¹⁶³ [i]