MinT - Best Known (235−118, 235, s)-Nets in Base 2

Best Known (235−118, 235, s)-Nets in Base 2

(235−118, 235, 57)-Net over F₂ — Constructive and digital

Digital (117, 235, 57)-net over F₂, using

t-expansion [i] based on digital (110, 235, 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]

(235−118, 235, 73)-Net over F₂ — Digital

Digital (117, 235, 73)-net over F₂, using

t-expansion [i] based on digital (114, 235, 73)-net over F₂, using
- net from sequence [i] based on digital (114, 72)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 114 and N(F) ≥ 73, using
    - Drinfeld modules of rank 1 [i]

(235−118, 235, 247)-Net over F₂ — Upper bound on s (digital)

There is no digital (117, 235, 248)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2²³⁵, 248, F₂, 118) (dual of [248, 13, 119]-code), but
- residual code [i] would yield OA(2¹¹⁷, 129, S₂, 59), but
  - 1 times truncation [i] would yield OA(2¹¹⁶, 128, S₂, 58), but
    - the linear programming bound shows that MÂ â‰¥ 156 848903 502620 073002 649233 113080 659968 / 1575 > 2¹¹⁶ [i]

(235−118, 235, 248)-Net in Base 2 — Upper bound on s

There is no (117, 235, 249)-net in base 2, because

12 times m-reduction [i] would yield (117, 223, 249)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²²³, 249, S₂, 106), but
  - the linear programming bound shows that MÂ â‰¥ 125060 361585 465082 046951 669439 069378 009061 149491 060510 099746 675837 762518 646784 / 7177 430007 > 2²²³ [i]