MinT - Best Known (120, 120+28, s)-Nets in Base 3

Best Known (120, 120+28, s)-Nets in Base 3

(120, 120+28, 688)-Net over F₃ — Constructive and digital

Digital (120, 148, 688)-net over F₃, using

t-expansion [i] based on digital (118, 148, 688)-net over F₃, using
- trace code for nets [i] based on digital (7, 37, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(120, 120+28, 3088)-Net over F₃ — Digital

Digital (120, 148, 3088)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁴⁸, 3088, F₃, 2, 28) (dual of [(3088, 2), 6028, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3¹⁴⁸, 3287, F₃, 2, 28) (dual of [(3287, 2), 6426, 29]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(3¹⁴⁸, 6574, F₃, 28) (dual of [6574, 6426, 29]-code), using
    - constructionÂ X applied to C_e(27) âŠ‚ C_e(24) [i] based on
      1. linear OA(3¹⁴⁵, 6561, F₃, 28) (dual of [6561, 6416, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
      2. linear OA(3¹²⁹, 6561, F₃, 25) (dual of [6561, 6432, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]
      3. linear OA(3³, 13, F₃, 2) (dual of [13, 10, 3]-code), using
        Hamming code H(3,3) [i]

(120, 120+28, 334397)-Net in Base 3 — Upper bound on s

There is no (120, 148, 334398)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 41110 090926 776859 341980 035766 034134 798750 824872 772546 189603 854514 703029 > 3¹⁴⁸ [i]