MinT - Best Known (129, 129+126, s)-Nets in Base 2

Best Known (129, 129+126, s)-Nets in Base 2

(129, 129+126, 57)-Net over F₂ — Constructive and digital

Digital (129, 255, 57)-net over F₂, using

t-expansion [i] based on digital (110, 255, 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]

(129, 129+126, 81)-Net over F₂ — Digital

Digital (129, 255, 81)-net over F₂, using

t-expansion [i] based on digital (126, 255, 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]

(129, 129+126, 273)-Net over F₂ — Upper bound on s (digital)

There is no digital (129, 255, 274)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2²⁵⁵, 274, F₂, 126) (dual of [274, 19, 127]-code), but
- residual code [i] would yield OA(2¹²⁹, 147, S₂, 63), but
  - 1 times truncation [i] would yield OA(2¹²⁸, 146, S₂, 62), but
    - the linear programming bound shows that MÂ â‰¥ 10 546031 115613 724859 656905 833525 360409 444352 / 30229 > 2¹²⁸ [i]

(129, 129+126, 284)-Net in Base 2 — Upper bound on s

There is no (129, 255, 285)-net in base 2, because

18 times m-reduction [i] would yield (129, 237, 285)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²³⁷, 285, S₂, 108), but
  - the linear programming bound shows that MÂ â‰¥ 56343 399996 195265 169227 114420 218575 369487 305782 122681 969429 903696 483971 902670 896111 812608 / 204485 216198 046875 > 2²³⁷ [i]