MinT - Best Known (170−15, 170, s)-Nets in Base 2

Best Known (170−15, 170, s)-Nets in Base 2

(170−15, 170, 1198376)-Net over F₂ — Constructive and digital

Digital (155, 170, 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, 162, 1198371)-net over F₂, using
  - net defined by OOA [i] based on linear OOA(2¹⁶², 1198371, F₂, 15, 15) (dual of [(1198371, 15), 17975403, 16]-NRT-code), using
    - OOA 7-folding and stacking with additional row [i] based on linear OA(2¹⁶², 8388598, F₂, 15) (dual of [8388598, 8388436, 16]-code), using
      - discarding factors / shortening the dual code based on linear OA(2¹⁶², large, F₂, 15) (dual of [large, large−162, 16]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 8388607Â = 2²³âˆ’1, defining interval IÂ =Â [0,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(170−15, 170, 1398105)-Net over F₂ — Digital

Digital (155, 170, 1398105)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁷⁰, 1398105, F₂, 6, 15) (dual of [(1398105, 6), 8388460, 16]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(2⁸, 5, F₂, 6, 7) (dual of [(5, 6), 22, 8]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(6;F,22P) [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(2¹⁶², 1398100, F₂, 6, 15) (dual of [(1398100, 6), 8388438, 16]-NRT-code), using
    - OOA 6-folding [i] based on linear OA(2¹⁶², 8388600, F₂, 15) (dual of [8388600, 8388438, 16]-code), using
      - discarding factors / shortening the dual code based on linear OA(2¹⁶², large, F₂, 15) (dual of [large, large−162, 16]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 8388607Â = 2²³âˆ’1, defining interval IÂ =Â [0,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(170−15, 170, large)-Net in Base 2 — Upper bound on s

There is no (155, 170, large)-net in base 2, because

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