MinT - Best Known (79, 79+73, s)-Nets in Base 2

Best Known (79, 79+73, s)-Nets in Base 2

(79, 79+73, 50)-Net over F₂ — Constructive and digital

Digital (79, 152, 50)-net over F₂, using

t-expansion [i] based on digital (75, 152, 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]

(79, 79+73, 52)-Net over F₂ — Digital

Digital (79, 152, 52)-net over F₂, using

t-expansion [i] based on digital (77, 152, 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]

(79, 79+73, 177)-Net over F₂ — Upper bound on s (digital)

There is no digital (79, 152, 178)-net over F₂, because

3 times m-reduction [i] would yield digital (79, 149, 178)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁴⁹, 178, F₂, 70) (dual of [178, 29, 71]-code), but
  - adding a parity check bit [i] would yield linear OA(2¹⁵⁰, 179, F₂, 71) (dual of [179, 29, 72]-code), but
    - â€œBKâ€ bound on codes from Brouwerâ€™s database [i]

(79, 79+73, 212)-Net in Base 2 — Upper bound on s

There is no (79, 152, 213)-net in base 2, because

1 times m-reduction [i] would yield (79, 151, 213)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3080 395052 153227 699468 482094 657997 669037 505470 > 2¹⁵¹ [i]