MinT - Best Known (174, 192, s)-Nets in Base 3

Best Known (174, 192, s)-Nets in Base 3

(174, 192, 932077)-Net over F₃ — Constructive and digital

Digital (174, 192, 932077)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (3, 12, 10)-net over F₃, using
  - net from sequence [i] based on digital (3, 9)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 3 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (162, 180, 932067)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁸⁰, 932067, F₃, 18, 18) (dual of [(932067, 18), 16777026, 19]-NRT-code), using
    - OA 9-folding and stacking [i] based on linear OA(3¹⁸⁰, large, F₃, 18) (dual of [large, large−180, 19]-code), using
      - the primitive narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(174, 192, 3550347)-Net over F₃ — Digital

Digital (174, 192, 3550347)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁹², 3550347, F₃, 2, 18) (dual of [(3550347, 2), 7100502, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3¹⁹², 4194311, F₃, 2, 18) (dual of [(4194311, 2), 8388430, 19]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(3¹², 10, F₃, 2, 9) (dual of [(10, 2), 8, 10]-NRT-code), using
      - extended algebraic-geometric NRT-code AG_e(2;F,10P) [i] based on function field F/F₃ with g(F) = 3 and N(F) ≥ 10, using
        an explicitly constructive algebraic function field [i]
    2. linear OOA(3¹⁸⁰, 4194301, F₃, 2, 18) (dual of [(4194301, 2), 8388422, 19]-NRT-code), using
      - OOA 2-folding [i] based on linear OA(3¹⁸⁰, 8388602, F₃, 18) (dual of [8388602, 8388422, 19]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹⁸⁰, large, F₃, 18) (dual of [large, large−180, 19]-code), using
        the primitive narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(174, 192, large)-Net in Base 3 — Upper bound on s

There is no (174, 192, large)-net in base 3, because

16 times m-reduction [i] would yield (174, 176, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]