MinT - Best Known (238−121, 238, s)-Nets in Base 2

Best Known (238−121, 238, s)-Nets in Base 2

(238−121, 238, 57)-Net over F₂ — Constructive and digital

Digital (117, 238, 57)-net over F₂, using

t-expansion [i] based on digital (110, 238, 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]

(238−121, 238, 73)-Net over F₂ — Digital

Digital (117, 238, 73)-net over F₂, using

t-expansion [i] based on digital (114, 238, 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]

(238−121, 238, 246)-Net over F₂ — Upper bound on s (digital)

There is no digital (117, 238, 247)-net over F₂, because

1 times m-reduction [i] would yield digital (117, 237, 247)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2²³⁷, 247, F₂, 120) (dual of [247, 10, 121]-code), but
  - residual code [i] would yield linear OA(2¹¹⁷, 126, F₂, 60) (dual of [126, 9, 61]-code), but
    - residual code [i] would yield linear OA(2⁵⁷, 65, F₂, 30) (dual of [65, 8, 31]-code), but
      - 2 times truncation [i] would yield linear OA(2⁵⁵, 63, F₂, 28) (dual of [63, 8, 29]-code), but
        â€œBJVâ€ bound on codes from Brouwerâ€™s database [i]

(238−121, 238, 248)-Net in Base 2 — Upper bound on s

There is no (117, 238, 249)-net in base 2, because

15 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]