MinT - Best Known (186−101, 186, s)-Nets in Base 2

Best Known (186−101, 186, s)-Nets in Base 2

(186−101, 186, 52)-Net over F₂ — Constructive and digital

Digital (85, 186, 52)-net over F₂, using

net from sequence [i] based on digital (85, 51)-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 3 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]

(186−101, 186, 57)-Net over F₂ — Digital

Digital (85, 186, 57)-net over F₂, using

t-expansion [i] based on digital (83, 186, 57)-net over F₂, using
- net from sequence [i] based on digital (83, 56)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 83 and N(F) ≥ 57, using
    - an algebraic function field by Auer [i]

(186−101, 186, 181)-Net over F₂ — Upper bound on s (digital)

There is no digital (85, 186, 182)-net over F₂, because

13 times m-reduction [i] would yield digital (85, 173, 182)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁷³, 182, F₂, 88) (dual of [182, 9, 89]-code), but
  - residual code [i] would yield linear OA(2⁸⁵, 93, F₂, 44) (dual of [93, 8, 45]-code), but
    - residual code [i] would yield linear OA(2⁴¹, 48, F₂, 22) (dual of [48, 7, 23]-code), but
      - residual code [i] would yield linear OA(2¹⁹, 25, F₂, 11) (dual of [25, 6, 12]-code), but
        1 times truncation [i] would yield linear OA(2¹⁸, 24, F₂, 10) (dual of [24, 6, 11]-code), but
        residual code [i] would yield linear OA(2⁸, 13, F₂, 5) (dual of [13, 5, 6]-code), but
        1 times truncation [i] would yield linear OA(2⁷, 12, F₂, 4) (dual of [12, 5, 5]-code), but
        bound by Fontaine and Peterson [i]

(186−101, 186, 182)-Net in Base 2 — Upper bound on s

There is no (85, 186, 183)-net in base 2, because

9 times m-reduction [i] would yield (85, 177, 183)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2¹⁷⁷, 183, S₂, 92), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 6 129982 163463 555433 433388 108601 236734 474956 488734 408704 / 31 > 2¹⁷⁷ [i]