MinT - Best Known (127, 127+122, s)-Nets in Base 2

Best Known (127, 127+122, s)-Nets in Base 2

(127, 127+122, 57)-Net over F₂ — Constructive and digital

Digital (127, 249, 57)-net over F₂, using

t-expansion [i] based on digital (110, 249, 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]

(127, 127+122, 81)-Net over F₂ — Digital

Digital (127, 249, 81)-net over F₂, using

t-expansion [i] based on digital (126, 249, 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]

(127, 127+122, 272)-Net over F₂ — Upper bound on s (digital)

There is no digital (127, 249, 273)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2²⁴⁹, 273, F₂, 122) (dual of [273, 24, 123]-code), but
- residual code [i] would yield OA(2¹²⁷, 150, S₂, 61), but
  - 1 times truncation [i] would yield OA(2¹²⁶, 149, S₂, 60), but
    - the linear programming bound shows that MÂ â‰¥ 40 354766 457887 934259 048521 444548 255732 989952 / 438495 > 2¹²⁶ [i]

(127, 127+122, 277)-Net in Base 2 — Upper bound on s

There is no (127, 249, 278)-net in base 2, because

14 times m-reduction [i] would yield (127, 235, 278)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²³⁵, 278, S₂, 108), but
  - the linear programming bound shows that MÂ â‰¥ 8610 120597 997849 103535 796060 969471 925725 360797 699038 214930 323792 633970 252546 589383 983104 / 113493 110947 734375 > 2²³⁵ [i]