MinT - Best Known (54−11, 54, s)-Nets in Base 3

Best Known (54−11, 54, s)-Nets in Base 3

(54−11, 54, 464)-Net over F₃ — Constructive and digital

Digital (43, 54, 464)-net over F₃, using

3² times duplication [i] based on digital (41, 52, 464)-net over F₃, using
- trace code for nets [i] based on digital (2, 13, 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]

(54−11, 54, 1330)-Net over F₃ — Digital

Digital (43, 54, 1330)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁵⁴, 1330, F₃, 11) (dual of [1330, 1276, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁵⁴, 2205, F₃, 11) (dual of [2205, 2151, 12]-code), using
  - constructionÂ X applied to C_e(10) âŠ‚ C_e(7) [i] based on
    1. 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]
    2. linear OA(3³⁶, 2187, F₃, 8) (dual of [2187, 2151, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [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]

(54−11, 54, 148688)-Net in Base 3 — Upper bound on s

There is no (43, 54, 148689)-net in base 3, because

1 times m-reduction [i] would yield (43, 53, 148689)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 19 383341 717503 509735 293587 > 3⁵³ [i]