MinT - Best Known (182−37, 182, s)-Nets in Base 3

Best Known (182−37, 182, s)-Nets in Base 3

(182−37, 182, 688)-Net over F₃ — Constructive and digital

Digital (145, 182, 688)-net over F₃, using

2 times m-reduction [i] based on digital (145, 184, 688)-net over F₃, using
- trace code for nets [i] based on digital (7, 46, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(182−37, 182, 2007)-Net over F₃ — Digital

Digital (145, 182, 2007)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁸², 2007, F₃, 37) (dual of [2007, 1825, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁸², 2229, F₃, 37) (dual of [2229, 2047, 38]-code), using
  - 3 times code embedding in larger space [i] based on linear OA(3¹⁷⁹, 2226, F₃, 37) (dual of [2226, 2047, 38]-code), using
    - constructionÂ X applied to C([0,18]) âŠ‚ C([0,15]) [i] based on
      1. linear OA(3¹⁶⁹, 2188, F₃, 37) (dual of [2188, 2019, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,18], and minimum distance dÂ â‰¥ |{−18,−17,…,18}|+1Â =Â 38 (BCH-bound) [i]
      2. linear OA(3¹⁴¹, 2188, F₃, 31) (dual of [2188, 2047, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,15], and minimum distance dÂ â‰¥ |{−15,−14,…,15}|+1Â =Â 32 (BCH-bound) [i]
      3. linear OA(3¹⁰, 38, F₃, 5) (dual of [38, 28, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹⁰, 39, F₃, 5) (dual of [39, 29, 6]-code), using
        generator matrices provided on Edelâ€™s homepage [i]

(182−37, 182, 237021)-Net in Base 3 — Upper bound on s

There is no (145, 182, 237022)-net in base 3, because

1 times m-reduction [i] would yield (145, 181, 237022)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 228 545085 504818 415132 140444 545177 677527 103620 519330 959945 824524 452463 458592 632479 272541 > 3¹⁸¹ [i]