MinT - Best Known (131, 131+125, s)-Nets in Base 2

Best Known (131, 131+125, s)-Nets in Base 2

(131, 131+125, 59)-Net over F₂ — Constructive and digital

Digital (131, 256, 59)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (15, 77, 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, 179, 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]

(131, 131+125, 81)-Net over F₂ — Digital

Digital (131, 256, 81)-net over F₂, using

t-expansion [i] based on digital (126, 256, 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]

(131, 131+125, 287)-Net over F₂ — Upper bound on s (digital)

There is no digital (131, 256, 288)-net over F₂, because

1 times m-reduction [i] would yield digital (131, 255, 288)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2²⁵⁵, 288, F₂, 124) (dual of [288, 33, 125]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(2²⁵⁴, 278, F₂, 124) (dual of [278, 24, 125]-code), but
      - residual code [i] would yield linear OA(2¹³⁰, 153, F₂, 62) (dual of [153, 23, 63]-code), but
        adding a parity check bit [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, 288, 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]

(131, 131+125, 291)-Net in Base 2 — Upper bound on s

There is no (131, 256, 292)-net in base 2, because

17 times m-reduction [i] would yield (131, 239, 292)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²³⁹, 292, S₂, 108), but
  - the linear programming bound shows that MÂ â‰¥ 129 753212 264703 737764 759763 826530 014325 194921 459040 258227 297024 189374 782768 881401 211050 262528 / 138 951361 618522 578125 > 2²³⁹ [i]