MinT - Best Known (77−22, 77, s)-Nets in Base 7

Best Known (77−22, 77, s)-Nets in Base 7

(77−22, 77, 221)-Net over F₇ — Constructive and digital

Digital (55, 77, 221)-net over F₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 11, 21)-net over F₇, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 3, 8)-net over F₇, using
      - net from sequence [i] based on digital (0, 7)-sequence over F₇, using
        Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₇ with g(F) = 0 and N(F) ≥ 8, using
        the rational function field F₇(x) [i]
        Niederreiter sequence [i]
    2. digital (1, 8, 13)-net over F₇, using
      - 5 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
        a shift-net [i]
2. digital (11, 22, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 11, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄₉ with g(F) = 0 and N(F) ≥ 50, using
        the rational function field F₄₉(x) [i]
      - Niederreiter sequence [i]
3. digital (22, 44, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 22, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(77−22, 77, 2240)-Net over F₇ — Digital

Digital (55, 77, 2240)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7⁷⁷, 2240, F₇, 22) (dual of [2240, 2163, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(7⁷⁷, 2417, F₇, 22) (dual of [2417, 2340, 23]-code), using
  - constructionÂ X applied to C_e(21) âŠ‚ C_e(17) [i] based on
    1. linear OA(7⁷³, 2401, F₇, 22) (dual of [2401, 2328, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 7⁴âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
    2. linear OA(7⁶¹, 2401, F₇, 18) (dual of [2401, 2340, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 7⁴âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
    3. linear OA(7⁴, 16, F₇, 3) (dual of [16, 12, 4]-code or 16-cap in PG(3,7)), using
      - discarding factors / shortening the dual code based on linear OA(7⁴, 50, F₇, 3) (dual of [50, 46, 4]-code or 50-cap in PG(3,7)), using
        ovoid in PG(3,Â 7) [i]

(77−22, 77, 673821)-Net in Base 7 — Upper bound on s

There is no (55, 77, 673822)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 118182 598039 582389 345608 332188 887158 942222 962259 154703 337154 231353 > 7⁷⁷ [i]