MinT - Best Known (146−13, 146, s)-Nets in Base 2

Best Known (146−13, 146, s)-Nets in Base 2

(146−13, 146, 1398105)-Net over F₂ — Constructive and digital

Digital (133, 146, 1398105)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 7, 5)-net over F₂, using
  - net from sequence [i] based on digital (1, 4)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 1 and N(F) ≥ 5, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (126, 139, 1398100)-net over F₂, using
  - net defined by OOA [i] based on linear OOA(2¹³⁹, 1398100, F₂, 13, 13) (dual of [(1398100, 13), 18175161, 14]-NRT-code), using
    - OOA 6-folding and stacking with additional row [i] based on linear OA(2¹³⁹, 8388601, F₂, 13) (dual of [8388601, 8388462, 14]-code), using
      - discarding factors / shortening the dual code based on linear OA(2¹³⁹, large, F₂, 13) (dual of [large, large−139, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 8388609Â | 2⁴⁶âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(146−13, 146, 1677725)-Net over F₂ — Digital

Digital (133, 146, 1677725)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁴⁶, 1677725, F₂, 5, 13) (dual of [(1677725, 5), 8388479, 14]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(2⁷, 5, F₂, 5, 6) (dual of [(5, 5), 18, 7]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(5;F,18P) [i] based on function field F/F₂ with g(F) = 1 and N(F) ≥ 5, using
      - an explicitly constructive algebraic function field [i]
  2. linear OOA(2¹³⁹, 1677720, F₂, 5, 13) (dual of [(1677720, 5), 8388461, 14]-NRT-code), using
    - OOA 5-folding [i] based on linear OA(2¹³⁹, 8388600, F₂, 13) (dual of [8388600, 8388461, 14]-code), using
      - discarding factors / shortening the dual code based on linear OA(2¹³⁹, large, F₂, 13) (dual of [large, large−139, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 8388609Â | 2⁴⁶âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(146−13, 146, large)-Net in Base 2 — Upper bound on s

There is no (133, 146, large)-net in base 2, because

11 times m-reduction [i] would yield (133, 135, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]