MinT - Best Known (198−99, 198, s)-Nets in Base 2

Best Known (198−99, 198, s)-Nets in Base 2

(198−99, 198, 54)-Net over F₂ — Constructive and digital

Digital (99, 198, 54)-net over F₂, using

t-expansion [i] based on digital (95, 198, 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]

(198−99, 198, 65)-Net over F₂ — Digital

Digital (99, 198, 65)-net over F₂, using

t-expansion [i] based on digital (95, 198, 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]

(198−99, 198, 213)-Net over F₂ — Upper bound on s (digital)

There is no digital (99, 198, 214)-net over F₂, because

1 times m-reduction [i] would yield digital (99, 197, 214)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁹⁷, 214, F₂, 98) (dual of [214, 17, 99]-code), but
  - residual code [i] would yield OA(2⁹⁹, 115, S₂, 49), but
    - 1 times truncation [i] would yield OA(2⁹⁸, 114, S₂, 48), but
      - the linear programming bound shows that MÂ â‰¥ 141449 524575 866749 536608 130468 675584 / 431375 > 2⁹⁸ [i]

(198−99, 198, 244)-Net in Base 2 — Upper bound on s

There is no (99, 198, 245)-net in base 2, because

1 times m-reduction [i] would yield (99, 197, 245)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 231299 670306 675673 975156 310271 796327 608106 253105 200143 281442 > 2¹⁹⁷ [i]