MinT - Best Known (19, 28, s)-Nets in Base 7

Best Known (19, 28, s)-Nets in Base 7

(19, 28, 202)-Net over F₇ — Constructive and digital

Digital (19, 28, 202)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (4, 8, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 4, 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]
2. digital (11, 20, 102)-net over F₇, using
  - trace code for nets [i] based on digital (1, 10, 51)-net over F₄₉, using
    - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
      - Niederreiter sequence [i]

(19, 28, 804)-Net over F₇ — Digital

Digital (19, 28, 804)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7²⁸, 804, F₇, 9) (dual of [804, 776, 10]-code), using
- constructionÂ X applied to C([130,138]) âŠ‚ C([131,138]) [i] based on
  1. linear OA(7²⁸, 800, F₇, 9) (dual of [800, 772, 10]-code), using the BCH-code C(I) with length 800Â | 7⁴âˆ’1, defining interval IÂ =Â {130,131,…,138}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
  2. linear OA(7²⁴, 800, F₇, 8) (dual of [800, 776, 9]-code), using the BCH-code C(I) with length 800Â | 7⁴âˆ’1, defining interval IÂ =Â {131,132,…,138}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
  3. linear OA(7⁰, 4, F₇, 0) (dual of [4, 4, 1]-code), using
    - discarding factors / shortening the dual code based on linear OA(7⁰, s, F₇, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
      - OA with only one run / dual of code without redundancy [i]

(19, 28, 186770)-Net in Base 7 — Upper bound on s

There is no (19, 28, 186771)-net in base 7, because

1 times m-reduction [i] would yield (19, 27, 186771)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 65713 059583 343592 792841 > 7²⁷ [i]