MinT - Best Known (127, 127+128, s)-Nets in Base 2

Best Known (127, 127+128, s)-Nets in Base 2

(127, 127+128, 57)-Net over F₂ — Constructive and digital

Digital (127, 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]

(127, 127+128, 81)-Net over F₂ — Digital

Digital (127, 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]

(127, 127+128, 264)-Net over F₂ — Upper bound on s (digital)

There is no digital (127, 255, 265)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2²⁵⁵, 265, F₂, 128) (dual of [265, 10, 129]-code), but
- residual code [i] would yield linear OA(2¹²⁷, 136, F₂, 64) (dual of [136, 9, 65]-code), but
  - residual code [i] would yield linear OA(2⁶³, 71, F₂, 32) (dual of [71, 8, 33]-code), but
    - residual code [i] would yield linear OA(2³¹, 38, F₂, 16) (dual of [38, 7, 17]-code), but
      - residual code [i] would yield linear OA(2¹⁵, 21, F₂, 8) (dual of [21, 6, 9]-code), but
        residual code [i] would yield linear OA(2⁷, 12, F₂, 4) (dual of [12, 5, 5]-code), but
        bound by Fontaine and Peterson [i]

(127, 127+128, 277)-Net in Base 2 — Upper bound on s

There is no (127, 255, 278)-net in base 2, because

20 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]