MinT - Best Known (196, 196+29, s)-Nets in Base 3

Best Known (196, 196+29, s)-Nets in Base 3

(196, 196+29, 12661)-Net over F₃ — Constructive and digital

Digital (196, 225, 12661)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (1, 15, 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 (181, 210, 12654)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3²¹⁰, 12654, F₃, 29, 29) (dual of [(12654, 29), 366756, 30]-NRT-code), using
    - OOA 14-folding and stacking with additional row [i] based on linear OA(3²¹⁰, 177157, F₃, 29) (dual of [177157, 176947, 30]-code), using
      - discarding factors / shortening the dual code based on linear OA(3²¹⁰, 177158, F₃, 29) (dual of [177158, 176948, 30]-code), using
        constructionÂ X applied to C_e(28) âŠ‚ C_e(27) [i] based on
        linear OA(3²¹⁰, 177147, F₃, 29) (dual of [177147, 176937, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 177146Â = 3¹¹âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
        linear OA(3¹⁹⁹, 177147, F₃, 28) (dual of [177147, 176948, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 177146Â = 3¹¹âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
        linear OA(3⁰, 11, F₃, 0) (dual of [11, 11, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(3⁰, s, F₃, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(196, 196+29, 65212)-Net over F₃ — Digital

Digital (196, 225, 65212)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²²⁵, 65212, F₃, 2, 29) (dual of [(65212, 2), 130199, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3²²⁵, 88608, F₃, 2, 29) (dual of [(88608, 2), 176991, 30]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(3²²⁵, 177216, F₃, 29) (dual of [177216, 176991, 30]-code), using
    - discarding factors / shortening the dual code based on linear OA(3²²⁵, 177217, F₃, 29) (dual of [177217, 176992, 30]-code), using
      - constructionÂ X applied to C_e(28) âŠ‚ C_e(21) [i] based on
        linear OA(3²¹⁰, 177147, F₃, 29) (dual of [177147, 176937, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 177146Â = 3¹¹âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
        linear OA(3¹⁵⁵, 177147, F₃, 22) (dual of [177147, 176992, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 177146Â = 3¹¹âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
        linear OA(3¹⁵, 70, F₃, 6) (dual of [70, 55, 7]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹⁵, 85, F₃, 6) (dual of [85, 70, 7]-code), using
        codes constructed by inverting constructionÂ Y₁ [i]

(196, 196+29, large)-Net in Base 3 — Upper bound on s

There is no (196, 225, large)-net in base 3, because

27 times m-reduction [i] would yield (196, 198, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (196, 196+29, s)-Nets in Base 3

(196, 196+29, 12661)-Net over F3 — Constructive and digital

(196, 196+29, 65212)-Net over F3 — Digital

(196, 196+29, large)-Net in Base 3 — Upper bound on s

(196, 196+29, 12661)-Net over F₃ — Constructive and digital

(196, 196+29, 65212)-Net over F₃ — Digital