MinT - Best Known (99, 99+92, s)-Nets in Base 2

Best Known (99, 99+92, s)-Nets in Base 2

(99, 99+92, 54)-Net over F₂ — Constructive and digital

Digital (99, 191, 54)-net over F₂, using

t-expansion [i] based on digital (95, 191, 54)-net over F₂, using
- net from sequence [i] based on digital (95, 53)-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 5 places 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]

(99, 99+92, 65)-Net over F₂ — Digital

Digital (99, 191, 65)-net over F₂, using

t-expansion [i] based on digital (95, 191, 65)-net over F₂, using
- net from sequence [i] based on digital (95, 64)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 95 and N(F) ≥ 65, using
    - Drinfeld modules of rank 1 [i]

(99, 99+92, 216)-Net over F₂ — Upper bound on s (digital)

There is no digital (99, 191, 217)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2¹⁹¹, 217, F₂, 92) (dual of [217, 26, 93]-code), but
- 3 times code embedding in larger space [i] would yield linear OA(2¹⁹⁴, 220, F₂, 92) (dual of [220, 26, 93]-code), but
  - adding a parity check bit [i] would yield linear OA(2¹⁹⁵, 221, F₂, 93) (dual of [221, 26, 94]-code), but
    - â€œBKâ€ bound on codes from Brouwerâ€™s database [i]

(99, 99+92, 257)-Net in Base 2 — Upper bound on s

There is no (99, 191, 258)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3368 234897 967805 420265 983342 052434 376150 218094 543188 488064 > 2¹⁹¹ [i]