MinT - Best Known (53, 53+17, s)-Nets in Base 3

Best Known (53, 53+17, s)-Nets in Base 3

(53, 53+17, 328)-Net over F₃ — Constructive and digital

Digital (53, 70, 328)-net over F₃, using

3² times duplication [i] based on digital (51, 68, 328)-net over F₃, using
- trace code for nets [i] based on digital (0, 17, 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]

(53, 53+17, 490)-Net over F₃ — Digital

Digital (53, 70, 490)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁷⁰, 490, F₃, 17) (dual of [490, 420, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁷⁰, 742, F₃, 17) (dual of [742, 672, 18]-code), using
  - constructionÂ X applied to C_e(16) âŠ‚ C_e(13) [i] based on
    1. linear OA(3⁶⁷, 729, F₃, 17) (dual of [729, 662, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
    2. 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]
    3. linear OA(3³, 13, F₃, 2) (dual of [13, 10, 3]-code), using
      - Hamming code H(3,3) [i]

(53, 53+17, 24530)-Net in Base 3 — Upper bound on s

There is no (53, 70, 24531)-net in base 3, because

1 times m-reduction [i] would yield (53, 69, 24531)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 834 648077 297580 502650 042509 564145 > 3⁶⁹ [i]