MinT - Best Known (200, 219, s)-Nets in Base 2

Best Known (200, 219, s)-Nets in Base 2

(200, 219, 932072)-Net over F₂ — Constructive and digital

Digital (200, 219, 932072)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (2, 11, 6)-net over F₂, using
  - net from sequence [i] based on digital (2, 5)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 2 and N(F) ≥ 6, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (189, 208, 932066)-net over F₂, using
  - net defined by OOA [i] based on linear OOA(2²⁰⁸, 932066, F₂, 19, 19) (dual of [(932066, 19), 17709046, 20]-NRT-code), using
    - OOA 9-folding and stacking with additional row [i] based on linear OA(2²⁰⁸, 8388595, F₂, 19) (dual of [8388595, 8388387, 20]-code), using
      - discarding factors / shortening the dual code based on linear OA(2²⁰⁸, large, F₂, 19) (dual of [large, large−208, 20]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 8388607Â = 2²³âˆ’1, defining interval IÂ =Â [0,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]

(200, 219, 1198377)-Net over F₂ — Digital

Digital (200, 219, 1198377)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²¹⁹, 1198377, F₂, 7, 19) (dual of [(1198377, 7), 8388420, 20]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(2¹¹, 6, F₂, 7, 9) (dual of [(6, 7), 31, 10]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(7;F,32P) [i] based on function field F/F₂ with g(F) = 2 and N(F) ≥ 6, using
      - an explicitly constructive algebraic function field [i]
  2. linear OOA(2²⁰⁸, 1198371, F₂, 7, 19) (dual of [(1198371, 7), 8388389, 20]-NRT-code), using
    - OOA 7-folding [i] based on linear OA(2²⁰⁸, 8388597, F₂, 19) (dual of [8388597, 8388389, 20]-code), using
      - discarding factors / shortening the dual code based on linear OA(2²⁰⁸, large, F₂, 19) (dual of [large, large−208, 20]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 8388607Â = 2²³âˆ’1, defining interval IÂ =Â [0,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]

(200, 219, large)-Net in Base 2 — Upper bound on s

There is no (200, 219, large)-net in base 2, because

17 times m-reduction [i] would yield (200, 202, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]