MinT - Best Known (244−123, 244, s)-Nets in Base 2

Best Known (244−123, 244, s)-Nets in Base 2

(244−123, 244, 57)-Net over F₂ — Constructive and digital

Digital (121, 244, 57)-net over F₂, using

t-expansion [i] based on digital (110, 244, 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]

(244−123, 244, 80)-Net over F₂ — Digital

Digital (121, 244, 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]

(244−123, 244, 255)-Net over F₂ — Upper bound on s (digital)

There is no digital (121, 244, 256)-net over F₂, because

1 times m-reduction [i] would yield digital (121, 243, 256)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2²⁴³, 256, F₂, 122) (dual of [256, 13, 123]-code), but
  - residual code [i] would yield OA(2¹²¹, 133, S₂, 61), but
    - 1 times truncation [i] would yield OA(2¹²⁰, 132, S₂, 60), but
      - the linear programming bound shows that MÂ â‰¥ 16503 694795 665515 477973 668460 440758 255616 / 10323 > 2¹²⁰ [i]

(244−123, 244, 258)-Net in Base 2 — Upper bound on s

There is no (121, 244, 259)-net in base 2, because

15 times m-reduction [i] would yield (121, 229, 259)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²²⁹, 259, S₂, 108), but
  - the linear programming bound shows that MÂ â‰¥ 17498 495097 789040 453213 386774 622994 352540 479460 641070 835741 390745 701581 780259 176448 / 19 797109 545047 > 2²²⁹ [i]