MinT - Best Known (78, 78+20, s)-Nets in Base 3

Best Known (78, 78+20, s)-Nets in Base 3

(78, 78+20, 600)-Net over F₃ — Constructive and digital

Digital (78, 98, 600)-net over F₃, using

3² times duplication [i] based on digital (76, 96, 600)-net over F₃, using
- trace code for nets [i] based on digital (4, 24, 150)-net over F₈₁, using
  - net from sequence [i] based on digital (4, 149)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 4 and N(F) ≥ 150, using
      - a function field by SÃ©mirat [i]

(78, 78+20, 1390)-Net over F₃ — Digital

Digital (78, 98, 1390)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁹⁸, 1390, F₃, 20) (dual of [1390, 1292, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁹⁸, 2214, F₃, 20) (dual of [2214, 2116, 21]-code), using
  - constructionÂ X applied to C_e(19) âŠ‚ C_e(15) [i] based on
    1. linear OA(3⁹², 2187, F₃, 20) (dual of [2187, 2095, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
    2. linear OA(3⁷¹, 2187, F₃, 16) (dual of [2187, 2116, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
    3. linear OA(3⁶, 27, F₃, 3) (dual of [27, 21, 4]-code or 27-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]

(78, 78+20, 107323)-Net in Base 3 — Upper bound on s

There is no (78, 98, 107324)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 57265 073562 095634 002399 125674 369711 029956 656521 > 3⁹⁸ [i]