MinT - Best Known (169−14, 169, s)-Nets in Base 2

Best Known (169−14, 169, s)-Nets in Base 2

(169−14, 169, 1198376)-Net over F₂ — Constructive and digital

Digital (155, 169, 1198376)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 8, 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 (147, 161, 1198371)-net over F₂, using
  - net defined by OOA [i] based on linear OOA(2¹⁶¹, 1198371, F₂, 14, 14) (dual of [(1198371, 14), 16777033, 15]-NRT-code), using
    - OA 7-folding and stacking [i] based on linear OA(2¹⁶¹, 8388597, F₂, 14) (dual of [8388597, 8388436, 15]-code), using
      - discarding factors / shortening the dual code based on linear OA(2¹⁶¹, large, F₂, 14) (dual of [large, large−161, 15]-code), using
        the primitive narrow-sense BCH-code C(I) with length 8388607Â = 2²³âˆ’1, defining interval IÂ =Â [1,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(169−14, 169, 1677725)-Net over F₂ — Digital

Digital (155, 169, 1677725)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁶⁹, 1677725, F₂, 5, 14) (dual of [(1677725, 5), 8388456, 15]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(2⁸, 5, F₂, 5, 7) (dual of [(5, 5), 17, 8]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(5;F,17P) [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(2¹⁶¹, 1677720, F₂, 5, 14) (dual of [(1677720, 5), 8388439, 15]-NRT-code), using
    - OOA 5-folding [i] based on linear OA(2¹⁶¹, 8388600, F₂, 14) (dual of [8388600, 8388439, 15]-code), using
      - discarding factors / shortening the dual code based on linear OA(2¹⁶¹, large, F₂, 14) (dual of [large, large−161, 15]-code), using
        the primitive narrow-sense BCH-code C(I) with length 8388607Â = 2²³âˆ’1, defining interval IÂ =Â [1,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(169−14, 169, large)-Net in Base 2 — Upper bound on s

There is no (155, 169, large)-net in base 2, because

12 times m-reduction [i] would yield (155, 157, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]