MinT - Best Known (77, 151, s)-Nets in Base 2

Best Known (77, 151, s)-Nets in Base 2

(77, 151, 50)-Net over F₂ — Constructive and digital

Digital (77, 151, 50)-net over F₂, using

t-expansion [i] based on digital (75, 151, 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]

(77, 151, 52)-Net over F₂ — Digital

Digital (77, 151, 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]

(77, 151, 173)-Net over F₂ — Upper bound on s (digital)

There is no digital (77, 151, 174)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2¹⁵¹, 174, F₂, 74) (dual of [174, 23, 75]-code), but
- 4 times code embedding in larger space [i] would yield linear OA(2¹⁵⁵, 178, F₂, 74) (dual of [178, 23, 75]-code), but
  - adding a parity check bit [i] would yield linear OA(2¹⁵⁶, 179, F₂, 75) (dual of [179, 23, 76]-code), but
    - â€œBKâ€ bound on codes from Brouwerâ€™s database [i]

(77, 151, 197)-Net in Base 2 — Upper bound on s

There is no (77, 151, 198)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 2866 983943 394889 219150 889882 729542 875940 914742 > 2¹⁵¹ [i]