MinT - Best Known (83−20, 83, s)-Nets in Base 3

Best Known (83−20, 83, s)-Nets in Base 3

(83−20, 83, 328)-Net over F₃ — Constructive and digital

Digital (63, 83, 328)-net over F₃, using

1 times m-reduction [i] based on digital (63, 84, 328)-net over F₃, using
- trace code for nets [i] based on digital (0, 21, 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]

(83−20, 83, 547)-Net over F₃ — Digital

Digital (63, 83, 547)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁸³, 547, F₃, 20) (dual of [547, 464, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁸³, 745, F₃, 20) (dual of [745, 662, 21]-code), using
  - constructionÂ X applied to C_e(19) âŠ‚ C_e(16) [i] based on
    1. linear OA(3⁷⁹, 729, F₃, 20) (dual of [729, 650, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
    2. 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]
    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]

(83−20, 83, 20646)-Net in Base 3 — Upper bound on s

There is no (63, 83, 20647)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3991 203867 888517 211508 246140 285537 383101 > 3⁸³ [i]