MinT - Best Known (52, 71, s)-Nets in Base 3

Best Known (52, 71, s)-Nets in Base 3

(52, 71, 192)-Net over F₃ — Constructive and digital

Digital (52, 71, 192)-net over F₃, using

1 times m-reduction [i] based on digital (52, 72, 192)-net over F₃, using
- trace code for nets [i] based on digital (4, 24, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 4 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]

(52, 71, 317)-Net over F₃ — Digital

Digital (52, 71, 317)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁷¹, 317, F₃, 19) (dual of [317, 246, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁷¹, 372, F₃, 19) (dual of [372, 301, 20]-code), using
  - constructionÂ X applied to C_e(18) âŠ‚ C_e(16) [i] based on
    1. linear OA(3⁷⁰, 365, F₃, 19) (dual of [365, 295, 20]-code), using an extension C_e(18) of the narrow-sense BCH-code C(I) with length 364Â | 3⁶âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
    2. linear OA(3⁶⁴, 365, F₃, 17) (dual of [365, 301, 18]-code), using an extension C_e(16) of the narrow-sense BCH-code C(I) with length 364Â | 3⁶âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
    3. linear OA(3¹, 7, F₃, 1) (dual of [7, 6, 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]

(52, 71, 10649)-Net in Base 3 — Upper bound on s

There is no (52, 71, 10650)-net in base 3, because

1 times m-reduction [i] would yield (52, 70, 10650)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2504 786365 975619 488108 206875 324261 > 3⁷⁰ [i]