MinT - Best Known (197, 197+45, s)-Nets in Base 3

Best Known (197, 197+45, s)-Nets in Base 3

(197, 197+45, 896)-Net over F₃ — Constructive and digital

Digital (197, 242, 896)-net over F₃, using

t-expansion [i] based on digital (196, 242, 896)-net over F₃, using
- 2 times m-reduction [i] based on digital (196, 244, 896)-net over F₃, using
  - trace code for nets [i] based on digital (13, 61, 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]

(197, 197+45, 3945)-Net over F₃ — Digital

Digital (197, 242, 3945)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²⁴², 3945, F₃, 45) (dual of [3945, 3703, 46]-code), using
- discarding factors / shortening the dual code based on linear OA(3²⁴², 6579, F₃, 45) (dual of [6579, 6337, 46]-code), using
  - constructionÂ X applied to C([0,22]) âŠ‚ C([0,21]) [i] based on
    1. linear OA(3²⁴¹, 6562, F₃, 45) (dual of [6562, 6321, 46]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,22], and minimum distance dÂ â‰¥ |{−22,−21,…,22}|+1Â =Â 46 (BCH-bound) [i]
    2. linear OA(3²²⁵, 6562, F₃, 43) (dual of [6562, 6337, 44]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,21], and minimum distance dÂ â‰¥ |{−21,−20,…,21}|+1Â =Â 44 (BCH-bound) [i]
    3. linear OA(3¹, 17, F₃, 1) (dual of [17, 16, 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]

(197, 197+45, 762881)-Net in Base 3 — Upper bound on s

There is no (197, 242, 762882)-net in base 3, because

1 times m-reduction [i] would yield (197, 241, 762882)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 9 687788 975890 457782 188143 021106 239161 358431 093224 392173 315881 906648 425615 907800 823991 315448 221786 056410 467309 733597 > 3²⁴¹ [i]