MinT - Best Known (54, 68, s)-Nets in Base 3

Best Known (54, 68, s)-Nets in Base 3

(54, 68, 464)-Net over F₃ — Constructive and digital

Digital (54, 68, 464)-net over F₃, using

t-expansion [i] based on digital (53, 68, 464)-net over F₃, using
- trace code for nets [i] based on digital (2, 17, 116)-net over F₈₁, using
  - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
      - a function field by SÃ©mirat [i]

(54, 68, 1209)-Net over F₃ — Digital

Digital (54, 68, 1209)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁶⁸, 1209, F₃, 14) (dual of [1209, 1141, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁶⁸, 2205, F₃, 14) (dual of [2205, 2137, 15]-code), using
  - constructionÂ X applied to C_e(13) âŠ‚ C_e(10) [i] based on
    1. linear OA(3⁶⁴, 2187, F₃, 14) (dual of [2187, 2123, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
    2. linear OA(3⁵⁰, 2187, F₃, 11) (dual of [2187, 2137, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
    3. linear OA(3⁴, 18, F₃, 2) (dual of [18, 14, 3]-code), using
      - discarding factors / shortening the dual code based on linear OA(3⁴, 40, F₃, 2) (dual of [40, 36, 3]-code), using
        Hamming code H(4,3) [i]

(54, 68, 72902)-Net in Base 3 — Upper bound on s

There is no (54, 68, 72903)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 278 143081 991662 735104 903716 515731 > 3⁶⁸ [i]