MinT - Best Known (253−127, 253, s)-Nets in Base 2

Best Known (253−127, 253, s)-Nets in Base 2

(253−127, 253, 57)-Net over F₂ — Constructive and digital

Digital (126, 253, 57)-net over F₂, using

t-expansion [i] based on digital (110, 253, 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]

(253−127, 253, 81)-Net over F₂ — Digital

Digital (126, 253, 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]

(253−127, 253, 266)-Net over F₂ — Upper bound on s (digital)

There is no digital (126, 253, 267)-net over F₂, because

3 times m-reduction [i] would yield digital (126, 250, 267)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2²⁵⁰, 267, F₂, 124) (dual of [267, 17, 125]-code), but
  - residual code [i] would yield OA(2¹²⁶, 142, S₂, 62), but
    - the linear programming bound shows that MÂ â‰¥ 944964 132939 446113 037791 284838 020323 213312 / 10545 > 2¹²⁶ [i]

(253−127, 253, 274)-Net in Base 2 — Upper bound on s

There is no (126, 253, 275)-net in base 2, because

19 times m-reduction [i] would yield (126, 234, 275)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²³⁴, 275, S₂, 108), but
  - the linear programming bound shows that MÂ â‰¥ 275 709011 416343 386283 194427 515396 341112 620119 523157 025502 536288 617777 395593 653660 942336 / 9364 570416 997125 > 2²³⁴ [i]