MinT - Best Known (194−33, 194, s)-Nets in Base 3

Best Known (194−33, 194, s)-Nets in Base 3

(194−33, 194, 896)-Net over F₃ — Constructive and digital

Digital (161, 194, 896)-net over F₃, using

t-expansion [i] based on digital (160, 194, 896)-net over F₃, using
- 2 times m-reduction [i] based on digital (160, 196, 896)-net over F₃, using
  - trace code for nets [i] based on digital (13, 49, 224)-net over F₈₁, using
    - net from sequence [i] based on digital (13, 223)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 13 and N(F) ≥ 224, using
        a function field by SÃ©mirat [i]

(194−33, 194, 5771)-Net over F₃ — Digital

Digital (161, 194, 5771)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁹⁴, 5771, F₃, 33) (dual of [5771, 5577, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁹⁴, 6596, F₃, 33) (dual of [6596, 6402, 34]-code), using
  - constructionÂ X4 applied to C([0,18]) âŠ‚ C([0,15]) [i] based on
    1. linear OA(3¹⁹³, 6562, F₃, 37) (dual of [6562, 6369, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,18], and minimum distance dÂ â‰¥ |{−18,−17,…,18}|+1Â =Â 38 (BCH-bound) [i]
    2. linear OA(3¹⁶¹, 6562, F₃, 31) (dual of [6562, 6401, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,15], and minimum distance dÂ â‰¥ |{−15,−14,…,15}|+1Â =Â 32 (BCH-bound) [i]
    3. linear OA(3³³, 34, F₃, 33) (dual of [34, 1, 34]-code or 34-arc in PG(32,3)), using
      - dual of repetition code with length 34 [i]
    4. linear OA(3¹, 34, F₃, 1) (dual of [34, 33, 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]

(194−33, 194, 1935443)-Net in Base 3 — Upper bound on s

There is no (161, 194, 1935444)-net in base 3, because

1 times m-reduction [i] would yield (161, 193, 1935444)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 121 452300 384489 696641 601578 854813 675081 370213 705504 583876 555045 784589 029414 760215 408987 616385 > 3¹⁹³ [i]