MinT - Best Known (58−12, 58, s)-Nets in Base 3

Best Known (58−12, 58, s)-Nets in Base 3

(58−12, 58, 464)-Net over F₃ — Constructive and digital

Digital (46, 58, 464)-net over F₃, using

3² times duplication [i] based on digital (44, 56, 464)-net over F₃, using
- trace code for nets [i] based on digital (2, 14, 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]

(58−12, 58, 1178)-Net over F₃ — Digital

Digital (46, 58, 1178)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁵⁸, 1178, F₃, 12) (dual of [1178, 1120, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁵⁸, 2202, F₃, 12) (dual of [2202, 2144, 13]-code), using
  - constructionÂ X applied to C_e(12) âŠ‚ C_e(9) [i] based on
    1. linear OA(3⁵⁷, 2187, F₃, 13) (dual of [2187, 2130, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
    2. linear OA(3⁴³, 2187, F₃, 10) (dual of [2187, 2144, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
    3. linear OA(3¹, 15, F₃, 1) (dual of [15, 14, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹, s, F₃, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(58−12, 58, 61280)-Net in Base 3 — Upper bound on s

There is no (46, 58, 61281)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 4710 167904 578180 088986 739865 > 3⁵⁸ [i]