MinT - Best Known (45, 45+14, s)-Nets in Base 3

Best Known (45, 45+14, s)-Nets in Base 3

(45, 45+14, 328)-Net over F₃ — Constructive and digital

Digital (45, 59, 328)-net over F₃, using

1 times m-reduction [i] based on digital (45, 60, 328)-net over F₃, using
- trace code for nets [i] based on digital (0, 15, 82)-net over F₈₁, using
  - net from sequence [i] based on digital (0, 81)-sequence over F₈₁, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 0 and N(F) ≥ 82, using
      - the rational function field F₈₁(x) [i]
    - Niederreiter sequence [i]

(45, 45+14, 525)-Net over F₃ — Digital

Digital (45, 59, 525)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁵⁹, 525, F₃, 14) (dual of [525, 466, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁵⁹, 745, F₃, 14) (dual of [745, 686, 15]-code), using
  - constructionÂ X applied to C_e(13) âŠ‚ C_e(10) [i] based on
    1. linear OA(3⁵⁵, 729, F₃, 14) (dual of [729, 674, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
    2. linear OA(3⁴³, 729, F₃, 11) (dual of [729, 686, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
    3. linear OA(3⁴, 16, F₃, 2) (dual of [16, 12, 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]

(45, 45+14, 17749)-Net in Base 3 — Upper bound on s

There is no (45, 59, 17750)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 14134 512099 704103 465357 392601 > 3⁵⁹ [i]