MinT - Best Known (82, 110, s)-Nets in Base 3

Best Known (82, 110, s)-Nets in Base 3

(82, 110, 252)-Net over F₃ — Constructive and digital

Digital (82, 110, 252)-net over F₃, using

1 times m-reduction [i] based on digital (82, 111, 252)-net over F₃, using
- trace code for nets [i] based on digital (8, 37, 84)-net over F₂₇, using
  - net from sequence [i] based on digital (8, 83)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 8 and N(F) ≥ 84, using
      - a function field by SÃ©mirat [i]

(82, 110, 506)-Net over F₃ — Digital

Digital (82, 110, 506)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹¹⁰, 506, F₃, 28) (dual of [506, 396, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹¹⁰, 736, F₃, 28) (dual of [736, 626, 29]-code), using
  - constructionÂ X applied to C_e(27) âŠ‚ C_e(25) [i] based on
    1. linear OA(3¹⁰⁹, 729, F₃, 28) (dual of [729, 620, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
    2. linear OA(3¹⁰³, 729, F₃, 26) (dual of [729, 626, 27]-code), using an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [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]

(82, 110, 16939)-Net in Base 3 — Upper bound on s

There is no (82, 110, 16940)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 30453 776826 413459 557286 075280 393233 443388 595956 912457 > 3¹¹⁰ [i]