MinT - Best Known (251−127, 251, s)-Nets in Base 2

Best Known (251−127, 251, s)-Nets in Base 2

(251−127, 251, 57)-Net over F₂ — Constructive and digital

Digital (124, 251, 57)-net over F₂, using

t-expansion [i] based on digital (110, 251, 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]

(251−127, 251, 80)-Net over F₂ — Digital

Digital (124, 251, 80)-net over F₂, using

t-expansion [i] based on digital (121, 251, 80)-net over F₂, using
- net from sequence [i] based on digital (121, 79)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 121 and N(F) ≥ 80, using
    - Drinfeld modules of rank 1 [i]

(251−127, 251, 260)-Net over F₂ — Upper bound on s (digital)

There is no digital (124, 251, 261)-net over F₂, because

3 times m-reduction [i] would yield digital (124, 248, 261)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2²⁴⁸, 261, F₂, 124) (dual of [261, 13, 125]-code), but
  - residual code [i] would yield OA(2¹²⁴, 136, S₂, 62), but
    - the linear programming bound shows that MÂ â‰¥ 17949 894855 079503 947693 010542 025773 154304 / 703 > 2¹²⁴ [i]

(251−127, 251, 267)-Net in Base 2 — Upper bound on s

There is no (124, 251, 268)-net in base 2, because

19 times m-reduction [i] would yield (124, 232, 268)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²³², 268, S₂, 108), but
  - the linear programming bound shows that MÂ â‰¥ 31 015917 364418 637531 270768 022329 527868 589587 272176 065660 880357 772788 956541 131711 053824 / 4226 244518 828475 > 2²³² [i]