MinT - Best Known (194−17, 194, s)-Nets in Base 2

Best Known (194−17, 194, s)-Nets in Base 2

(194−17, 194, 1048580)-Net over F₂ — Constructive and digital

Digital (177, 194, 1048580)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 9, 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 (168, 185, 1048575)-net over F₂, using
  - net defined by OOA [i] based on linear OOA(2¹⁸⁵, 1048575, F₂, 17, 17) (dual of [(1048575, 17), 17825590, 18]-NRT-code), using
    - OOA 8-folding and stacking with additional row [i] based on linear OA(2¹⁸⁵, 8388601, F₂, 17) (dual of [8388601, 8388416, 18]-code), using
      - discarding factors / shortening the dual code based on linear OA(2¹⁸⁵, large, F₂, 17) (dual of [large, large−185, 18]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 8388609Â | 2⁴⁶âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(194−17, 194, 1398105)-Net over F₂ — Digital

Digital (177, 194, 1398105)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁹⁴, 1398105, F₂, 6, 17) (dual of [(1398105, 6), 8388436, 18]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(2⁹, 5, F₂, 6, 8) (dual of [(5, 6), 21, 9]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(6;F,21P) [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(2¹⁸⁵, 1398100, F₂, 6, 17) (dual of [(1398100, 6), 8388415, 18]-NRT-code), using
    - OOA 6-folding [i] based on linear OA(2¹⁸⁵, 8388600, F₂, 17) (dual of [8388600, 8388415, 18]-code), using
      - discarding factors / shortening the dual code based on linear OA(2¹⁸⁵, large, F₂, 17) (dual of [large, large−185, 18]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 8388609Â | 2⁴⁶âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(194−17, 194, large)-Net in Base 2 — Upper bound on s

There is no (177, 194, large)-net in base 2, because

15 times m-reduction [i] would yield (177, 179, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]