MinT - Best Known (99, 196, s)-Nets in Base 2

Best Known (99, 196, s)-Nets in Base 2

(99, 196, 54)-Net over F₂ — Constructive and digital

Digital (99, 196, 54)-net over F₂, using

t-expansion [i] based on digital (95, 196, 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, 196, 65)-Net over F₂ — Digital

Digital (99, 196, 65)-net over F₂, using

t-expansion [i] based on digital (95, 196, 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, 196, 215)-Net over F₂ — Upper bound on s (digital)

There is no digital (99, 196, 216)-net over F₂, because

1 times m-reduction [i] would yield digital (99, 195, 216)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁹⁵, 216, F₂, 96) (dual of [216, 21, 97]-code), but
  - residual code [i] would yield OA(2⁹⁹, 119, S₂, 48), but
    - the linear programming bound shows that MÂ â‰¥ 7 222484 930221 945231 285920 393945 153536 / 10 168125 > 2⁹⁹ [i]

(99, 196, 248)-Net in Base 2 — Upper bound on s

There is no (99, 196, 249)-net in base 2, because

1 times m-reduction [i] would yield (99, 195, 249)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 56211 268133 255866 634800 016350 772202 522650 577072 332620 704933 > 2¹⁹⁵ [i]