MinT - Best Known (112−11, 112, s)-Nets in Base 3

Best Known (112−11, 112, s)-Nets in Base 3

(112−11, 112, 1677727)-Net over F₃ — Constructive and digital

Digital (101, 112, 1677727)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (1, 6, 7)-net over F₃, using
  - net from sequence [i] based on digital (1, 6)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 1 and N(F) ≥ 7, using
      - an explicitly constructive algebraic function field [i]
2. digital (95, 106, 1677720)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁰⁶, 1677720, F₃, 11, 11) (dual of [(1677720, 11), 18454814, 12]-NRT-code), using
    - OOA 5-folding and stacking with additional row [i] based on linear OA(3¹⁰⁶, 8388601, F₃, 11) (dual of [8388601, 8388495, 12]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁰⁶, large, F₃, 11) (dual of [large, large−106, 12]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(112−11, 112, 4194308)-Net over F₃ — Digital

Digital (101, 112, 4194308)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹¹², 4194308, F₃, 2, 11) (dual of [(4194308, 2), 8388504, 12]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(3⁶, 7, F₃, 2, 5) (dual of [(7, 2), 8, 6]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,8P) [i] based on function field F/F₃ with g(F) = 1 and N(F) ≥ 7, using
      - an explicitly constructive algebraic function field [i]
  2. linear OOA(3¹⁰⁶, 4194301, F₃, 2, 11) (dual of [(4194301, 2), 8388496, 12]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(3¹⁰⁶, 8388602, F₃, 11) (dual of [8388602, 8388496, 12]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁰⁶, large, F₃, 11) (dual of [large, large−106, 12]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(112−11, 112, large)-Net in Base 3 — Upper bound on s

There is no (101, 112, large)-net in base 3, because

9 times m-reduction [i] would yield (101, 103, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]