MinT - Best Known (172−15, 172, s)-Nets in Base 2

Best Known (172−15, 172, s)-Nets in Base 2

(172−15, 172, 1198378)-Net over F₂ — Constructive and digital

Digital (157, 172, 1198378)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 10, 7)-net over F₂, using
  - net from sequence [i] based on digital (3, 6)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 3 and N(F) ≥ 7, 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]

(172−15, 172, 1597816)-Net over F₂ — Digital

Digital (157, 172, 1597816)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁷², 1597816, F₂, 5, 15) (dual of [(1597816, 5), 7988908, 16]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2¹⁷², 1677727, F₂, 5, 15) (dual of [(1677727, 5), 8388463, 16]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(2¹⁰, 7, F₂, 5, 7) (dual of [(7, 5), 25, 8]-NRT-code), using
      - extended algebraic-geometric NRT-code AG_e(5;F,27P) [i] based on function field F/F₂ with g(F) = 3 and N(F) ≥ 7, using
        an explicitly constructive algebraic function field [i]
    2. linear OOA(2¹⁶², 1677720, F₂, 5, 15) (dual of [(1677720, 5), 8388438, 16]-NRT-code), using
      - OOA 5-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]

(172−15, 172, large)-Net in Base 2 — Upper bound on s

There is no (157, 172, large)-net in base 2, because

13 times m-reduction [i] would yield (157, 159, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]