MinT - Best Known (56, 56+17, s)-Nets in Base 3

Best Known (56, 56+17, s)-Nets in Base 3

(56, 56+17, 400)-Net over F₃ — Constructive and digital

Digital (56, 73, 400)-net over F₃, using

3¹ times duplication [i] based on digital (55, 72, 400)-net over F₃, using
- trace code for nets [i] based on digital (1, 18, 100)-net over F₈₁, using
  - net from sequence [i] based on digital (1, 99)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 1 and N(F) ≥ 100, using
      - a function field by SÃ©mirat [i]

(56, 56+17, 613)-Net over F₃ — Digital

Digital (56, 73, 613)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁷³, 613, F₃, 17) (dual of [613, 540, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁷³, 753, F₃, 17) (dual of [753, 680, 18]-code), using
  - constructionÂ X applied to C_e(16) âŠ‚ C_e(12) [i] based on
    1. linear OA(3⁶⁷, 729, F₃, 17) (dual of [729, 662, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
    2. linear OA(3⁴⁹, 729, F₃, 13) (dual of [729, 680, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
    3. linear OA(3⁶, 24, F₃, 3) (dual of [24, 18, 4]-code or 24-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]

(56, 56+17, 37039)-Net in Base 3 — Upper bound on s

There is no (56, 73, 37040)-net in base 3, because

1 times m-reduction [i] would yield (56, 72, 37040)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 22531 527525 205029 446965 309057 980417 > 3⁷² [i]