MinT - Best Known (258−126, 258, s)-Nets in Base 2

Best Known (258−126, 258, s)-Nets in Base 2

(258−126, 258, 59)-Net over F₂ — Constructive and digital

Digital (132, 258, 59)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (15, 78, 17)-net over F₂, using
  - net from sequence [i] based on digital (15, 16)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 15 and N(F) ≥ 17, using
      - an explicitly constructive algebraic function field [i]
2. digital (54, 180, 42)-net over F₂, using
  - net from sequence [i] based on digital (54, 41)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
      - an explicitly constructive algebraic function field [i]

(258−126, 258, 81)-Net over F₂ — Digital

Digital (132, 258, 81)-net over F₂, using

t-expansion [i] based on digital (126, 258, 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]

(258−126, 258, 290)-Net over F₂ — Upper bound on s (digital)

There is no digital (132, 258, 291)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2²⁵⁸, 291, F₂, 126) (dual of [291, 33, 127]-code), but
- constructionÂ Y1 [i] would yield
  1. linear OA(2²⁵⁷, 281, F₂, 126) (dual of [281, 24, 127]-code), but
    - residual code [i] would yield linear OA(2¹³¹, 154, F₂, 63) (dual of [154, 23, 64]-code), but
      - â€œBKâ€ bound on codes from Brouwerâ€™s database [i]
  2. OA(2³³, 291, S₂, 10), but
    - discarding factors would yield OA(2³³, 254, S₂, 10), but
      - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 8640 218941 > 2³³ [i]

(258−126, 258, 296)-Net in Base 2 — Upper bound on s

There is no (132, 258, 297)-net in base 2, because

18 times m-reduction [i] would yield (132, 240, 297)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²⁴⁰, 297, S₂, 108), but
  - the linear programming bound shows that MÂ â‰¥ 28020 398795 768772 015654 276848 365160 205592 563190 101250 876110 646415 347830 039205 258520 634937 311232 / 15527 302685 925212 739375 > 2²⁴⁰ [i]