MinT - Best Known (45, 45+12, s)-Nets in Base 3

Best Known (45, 45+12, s)-Nets in Base 3

(45, 45+12, 464)-Net over F₃ — Constructive and digital

Digital (45, 57, 464)-net over F₃, using

3¹ 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]

(45, 45+12, 1097)-Net over F₃ — Digital

Digital (45, 57, 1097)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3⁵⁷, 1097, F₃, 2, 12) (dual of [(1097, 2), 2137, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3⁵⁷, 2194, F₃, 12) (dual of [2194, 2137, 13]-code), using
  - 1 times truncation [i] based on linear OA(3⁵⁸, 2195, F₃, 13) (dual of [2195, 2137, 14]-code), using
    - constructionÂ X applied to C_e(12) âŠ‚ C_e(10) [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(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(3¹, 8, F₃, 1) (dual of [8, 7, 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]

(45, 45+12, 51026)-Net in Base 3 — Upper bound on s

There is no (45, 57, 51027)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1570 104398 673446 477642 490045 > 3⁵⁷ [i]