MinT - Best Known (179, 179+33, s)-Nets in Base 3

Best Known (179, 179+33, s)-Nets in Base 3

(179, 179+33, 1484)-Net over F₃ — Constructive and digital

Digital (179, 212, 1484)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (0, 16, 4)-net over F₃, using
  - net from sequence [i] based on digital (0, 3)-sequence over F₃, using
    - Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 0 and N(F) ≥ 4, using
      - the rational function field F₃(x) [i]
    - Niederreiter sequence [i]
2. digital (163, 196, 1480)-net over F₃, using
  - trace code for nets [i] based on digital (16, 49, 370)-net over F₈₁, using
    - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(179, 179+33, 10948)-Net over F₃ — Digital

Digital (179, 212, 10948)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²¹², 10948, F₃, 33) (dual of [10948, 10736, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3²¹², 19733, F₃, 33) (dual of [19733, 19521, 34]-code), using
  - 2 times code embedding in larger space [i] based on linear OA(3²¹⁰, 19731, F₃, 33) (dual of [19731, 19521, 34]-code), using
    - constructionÂ X applied to C([0,16]) âŠ‚ C([0,13]) [i] based on
      1. linear OA(3¹⁹⁹, 19684, F₃, 33) (dual of [19684, 19485, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684Â | 3¹⁸âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]
      2. linear OA(3¹⁶³, 19684, F₃, 27) (dual of [19684, 19521, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684Â | 3¹⁸âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (BCH-bound) [i]
      3. linear OA(3¹¹, 47, F₃, 5) (dual of [47, 36, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹¹, 85, F₃, 5) (dual of [85, 74, 6]-code), using
        codes constructed by inverting constructionÂ Y₁ [i]

(179, 179+33, 6661074)-Net in Base 3 — Upper bound on s

There is no (179, 212, 6661075)-net in base 3, because

1 times m-reduction [i] would yield (179, 211, 6661075)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 47052 822897 686758 088977 576887 991625 639422 975487 086739 731849 409354 548643 048394 999468 093808 947197 277921 > 3²¹¹ [i]