MinT - Best Known (130, 144, s)-Nets in Base 3

Best Known (130, 144, s)-Nets in Base 3

(130, 144, 1198378)-Net over F₃ — Constructive and digital

Digital (130, 144, 1198378)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (1, 8, 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 (122, 136, 1198371)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹³⁶, 1198371, F₃, 14, 14) (dual of [(1198371, 14), 16777058, 15]-NRT-code), using
    - OA 7-folding and stacking [i] based on linear OA(3¹³⁶, 8388597, F₃, 14) (dual of [8388597, 8388461, 15]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹³⁶, large, F₃, 14) (dual of [large, large−136, 15]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(130, 144, 3541478)-Net over F₃ — Digital

Digital (130, 144, 3541478)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁴⁴, 3541478, F₃, 2, 14) (dual of [(3541478, 2), 7082812, 15]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3¹⁴⁴, 4194308, F₃, 2, 14) (dual of [(4194308, 2), 8388472, 15]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(3⁸, 7, F₃, 2, 7) (dual of [(7, 2), 6, 8]-NRT-code), using
      - extended algebraic-geometric NRT-code AG_e(2;F,6P) [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(3¹³⁶, 4194301, F₃, 2, 14) (dual of [(4194301, 2), 8388466, 15]-NRT-code), using
      - OOA 2-folding [i] based on linear OA(3¹³⁶, 8388602, F₃, 14) (dual of [8388602, 8388466, 15]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹³⁶, large, F₃, 14) (dual of [large, large−136, 15]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(130, 144, large)-Net in Base 3 — Upper bound on s

There is no (130, 144, large)-net in base 3, because

12 times m-reduction [i] would yield (130, 132, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]