MinT - Best Known (253−125, 253, s)-Nets in Base 2

Best Known (253−125, 253, s)-Nets in Base 2

(253−125, 253, 57)-Net over F₂ — Constructive and digital

Digital (128, 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]

(253−125, 253, 81)-Net over F₂ — Digital

Digital (128, 253, 81)-net over F₂, using

t-expansion [i] based on 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]

(253−125, 253, 270)-Net over F₂ — Upper bound on s (digital)

There is no digital (128, 253, 271)-net over F₂, because

1 times m-reduction [i] would yield digital (128, 252, 271)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2²⁵², 271, F₂, 124) (dual of [271, 19, 125]-code), but
  - residual code [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]

(253−125, 253, 279)-Net in Base 2 — Upper bound on s

There is no (128, 253, 280)-net in base 2, because

17 times m-reduction [i] would yield (128, 236, 280)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²³⁶, 280, S₂, 108), but
  - the linear programming bound shows that MÂ â‰¥ 41504 278412 335844 454047 491725 650645 432186 567323 902384 277507 042637 995046 295745 131031 035904 / 340479 332843 203125 > 2²³⁶ [i]