MinT - Best Known (173, 191, s)-Nets in Base 3

Best Known (173, 191, s)-Nets in Base 3

(173, 191, 932075)-Net over F₃ — Constructive and digital

Digital (173, 191, 932075)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (2, 11, 8)-net over F₃, using
  - net from sequence [i] based on digital (2, 7)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 2 and N(F) ≥ 8, 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]

(173, 191, 3299610)-Net over F₃ — Digital

Digital (173, 191, 3299610)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁹¹, 3299610, F₃, 2, 18) (dual of [(3299610, 2), 6599029, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3¹⁹¹, 4194309, F₃, 2, 18) (dual of [(4194309, 2), 8388427, 19]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(3¹¹, 8, F₃, 2, 9) (dual of [(8, 2), 5, 10]-NRT-code), using
      - extended algebraic-geometric NRT-code AG_e(2;F,6P) [i] based on function field F/F₃ with g(F) = 2 and N(F) ≥ 8, 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]

(173, 191, large)-Net in Base 3 — Upper bound on s

There is no (173, 191, large)-net in base 3, because

16 times m-reduction [i] would yield (173, 175, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]