MinT - Best Known (65, 82, s)-Nets in Base 3

Best Known (65, 82, s)-Nets in Base 3

(65, 82, 464)-Net over F₃ — Constructive and digital

Digital (65, 82, 464)-net over F₃, using

2 times m-reduction [i] based on digital (65, 84, 464)-net over F₃, using
- trace code for nets [i] based on digital (2, 21, 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]

(65, 82, 1198)-Net over F₃ — Digital

Digital (65, 82, 1198)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁸², 1198, F₃, 17) (dual of [1198, 1116, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁸², 2205, F₃, 17) (dual of [2205, 2123, 18]-code), using
  - constructionÂ X applied to C_e(16) âŠ‚ C_e(13) [i] based on
    1. linear OA(3⁷⁸, 2187, F₃, 17) (dual of [2187, 2109, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
    2. linear OA(3⁶⁴, 2187, F₃, 14) (dual of [2187, 2123, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
    3. linear OA(3⁴, 18, F₃, 2) (dual of [18, 14, 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]

(65, 82, 127493)-Net in Base 3 — Upper bound on s

There is no (65, 82, 127494)-net in base 3, because

1 times m-reduction [i] would yield (65, 81, 127494)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 443 447069 798012 145626 117831 677009 600465 > 3⁸¹ [i]