MinT - Best Known (11, 18, s)-Nets in Base 7

Best Known (11, 18, s)-Nets in Base 7

(11, 18, 150)-Net over F₇ — Constructive and digital

Digital (11, 18, 150)-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 (7, 14, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 7, 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]

(11, 18, 322)-Net over F₇ — Digital

Digital (11, 18, 322)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7¹⁸, 322, F₇, 7) (dual of [322, 304, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(7¹⁸, 342, F₇, 7) (dual of [342, 324, 8]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 342Â = 7³âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]

(11, 18, 18624)-Net in Base 7 — Upper bound on s

There is no (11, 18, 18625)-net in base 7, because

1 times m-reduction [i] would yield (11, 17, 18625)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 232 646363 025751 > 7¹⁷ [i]