MinT - Best Known (78, 78+71, s)-Nets in Base 2

Best Known (78, 78+71, s)-Nets in Base 2

(78, 78+71, 50)-Net over F₂ — Constructive and digital

Digital (78, 149, 50)-net over F₂, using

t-expansion [i] based on digital (75, 149, 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, 78+71, 52)-Net over F₂ — Digital

Digital (78, 149, 52)-net over F₂, using

t-expansion [i] based on digital (77, 149, 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, 78+71, 175)-Net over F₂ — Upper bound on s (digital)

There is no digital (78, 149, 176)-net over F₂, because

1 times m-reduction [i] would yield digital (78, 148, 176)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁴⁸, 176, F₂, 70) (dual of [176, 28, 71]-code), but
  - adding a parity check bit [i] would yield linear OA(2¹⁴⁹, 177, F₂, 71) (dual of [177, 28, 72]-code), but
    - â€œBKâ€ bound on codes from Brouwerâ€™s database [i]

(78, 78+71, 213)-Net in Base 2 — Upper bound on s

There is no (78, 149, 214)-net in base 2, because

1 times m-reduction [i] would yield (78, 148, 214)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 408 441617 355578 995683 142262 066645 952028 781580 > 2¹⁴⁸ [i]