MinT - Best Known (259−129, 259, s)-Nets in Base 2

Best Known (259−129, 259, s)-Nets in Base 2

(259−129, 259, 57)-Net over F₂ — Constructive and digital

Digital (130, 259, 57)-net over F₂, using

t-expansion [i] based on digital (110, 259, 57)-net over F₂, using
- net from sequence [i] based on digital (110, 56)-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 8 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]

(259−129, 259, 81)-Net over F₂ — Digital

Digital (130, 259, 81)-net over F₂, using

t-expansion [i] based on digital (126, 259, 81)-net over F₂, using
- net from sequence [i] based on digital (126, 80)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 126 and N(F) ≥ 81, using
    - Drinfeld modules of rank 1 [i]

(259−129, 259, 274)-Net over F₂ — Upper bound on s (digital)

There is no digital (130, 259, 275)-net over F₂, because

1 times m-reduction [i] would yield digital (130, 258, 275)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2²⁵⁸, 275, F₂, 128) (dual of [275, 17, 129]-code), but
  - residual code [i] would yield OA(2¹³⁰, 146, S₂, 64), but
    - the linear programming bound shows that MÂ â‰¥ 4188 532932 170896 179262 970335 880775 460666 212352 / 2 937077 > 2¹³⁰ [i]

(259−129, 259, 286)-Net in Base 2 — Upper bound on s

There is no (130, 259, 287)-net in base 2, because

21 times m-reduction [i] would yield (130, 238, 287)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²³⁸, 287, S₂, 108), but
  - the linear programming bound shows that MÂ â‰¥ 2 133520 632200 595168 112459 587583 515002 010526 895398 363193 388655 973626 286863 730797 742512 930816 / 4 628313 007677 734375 > 2²³⁸ [i]