MinT - Best Known (78−22, 78, s)-Nets in Base 7

Best Known (78−22, 78, s)-Nets in Base 7

(78−22, 78, 226)-Net over F₇ — Constructive and digital

Digital (56, 78, 226)-net over F₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (5, 12, 26)-net over F₇, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 4, 50)-net over F₇, using
      - net defined by OOA [i] based on linear OOA(7⁴, 50, F₇, 3, 3) (dual of [(50, 3), 146, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(7⁴, 50, F₇, 2, 3) (dual of [(50, 2), 96, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] 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]
    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)

(78−22, 78, 2431)-Net over F₇ — Digital

Digital (56, 78, 2431)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7⁷⁸, 2431, F₇, 22) (dual of [2431, 2353, 23]-code), using
- 25 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 0, 1, 5 times 0, 1, 16 times 0) [i] based on 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]

(78−22, 78, 804215)-Net in Base 7 — Upper bound on s

There is no (56, 78, 804216)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 827272 197124 638814 921730 608994 013167 149696 311927 341607 923295 351553 > 7⁷⁸ [i]