MinT - Best Known (223−41, 223, s)-Nets in Base 3

Best Known (223−41, 223, s)-Nets in Base 3

(223−41, 223, 896)-Net over F₃ — Constructive and digital

Digital (182, 223, 896)-net over F₃, using

t-expansion [i] based on digital (181, 223, 896)-net over F₃, using
- 1 times m-reduction [i] based on digital (181, 224, 896)-net over F₃, using
  - trace code for nets [i] based on digital (13, 56, 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]

(223−41, 223, 3965)-Net over F₃ — Digital

Digital (182, 223, 3965)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²²³, 3965, F₃, 41) (dual of [3965, 3742, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(3²²³, 6591, F₃, 41) (dual of [6591, 6368, 42]-code), using
  - constructionÂ X applied to C_e(40) âŠ‚ C_e(36) [i] based on
    1. linear OA(3²¹⁷, 6561, F₃, 41) (dual of [6561, 6344, 42]-code), using an extension C_e(40) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,40], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 41 [i]
    2. linear OA(3¹⁹³, 6561, F₃, 37) (dual of [6561, 6368, 38]-code), using an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [i]
    3. linear OA(3⁶, 30, F₃, 3) (dual of [30, 24, 4]-code or 30-cap in PG(5,3)), using
      - discarding factors / shortening the dual code based on linear OA(3⁶, 48, F₃, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
        large caps in PG(5,Â b) [i]

(223−41, 223, 820940)-Net in Base 3 — Upper bound on s

There is no (182, 223, 820941)-net in base 3, because

1 times m-reduction [i] would yield (182, 222, 820941)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 8335 286474 868065 727229 128571 541888 036037 039021 973088 433978 201110 481355 531625 245882 806385 650690 744648 402513 > 3²²² [i]