MinT - Best Known (194−16, 194, s)-Nets in Base 2

Best Known (194−16, 194, s)-Nets in Base 2

(194−16, 194, 1048581)-Net over F₂ — Constructive and digital

Digital (178, 194, 1048581)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (2, 10, 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 (168, 184, 1048575)-net over F₂, using
  - net defined by OOA [i] based on linear OOA(2¹⁸⁴, 1048575, F₂, 16, 16) (dual of [(1048575, 16), 16777016, 17]-NRT-code), using
    - OA 8-folding and stacking [i] based on linear OA(2¹⁸⁴, 8388600, F₂, 16) (dual of [8388600, 8388416, 17]-code), using
      - discarding factors / shortening the dual code based on linear OA(2¹⁸⁴, large, F₂, 16) (dual of [large, large−184, 17]-code), using
        the primitive narrow-sense BCH-code C(I) with length 8388607Â = 2²³âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(194−16, 194, 1677726)-Net over F₂ — Digital

Digital (178, 194, 1677726)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁹⁴, 1677726, F₂, 5, 16) (dual of [(1677726, 5), 8388436, 17]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(2¹⁰, 6, F₂, 5, 8) (dual of [(6, 5), 20, 9]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(5;F,21P) [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(2¹⁸⁴, 1677720, F₂, 5, 16) (dual of [(1677720, 5), 8388416, 17]-NRT-code), using
    - OOA 5-folding [i] based on linear OA(2¹⁸⁴, 8388600, F₂, 16) (dual of [8388600, 8388416, 17]-code), using
      - discarding factors / shortening the dual code based on linear OA(2¹⁸⁴, large, F₂, 16) (dual of [large, large−184, 17]-code), using
        the primitive narrow-sense BCH-code C(I) with length 8388607Â = 2²³âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(194−16, 194, large)-Net in Base 2 — Upper bound on s

There is no (178, 194, large)-net in base 2, because

14 times m-reduction [i] would yield (178, 180, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]