MinT - Best Known (58, 81, s)-Nets in Base 7

Best Known (58, 81, s)-Nets in Base 7

(58, 81, 250)-Net over F₇ — Constructive and digital

Digital (58, 81, 250)-net over F₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (6, 13, 50)-net over F₇, using
  - base reduction for projective spaces (embedding PG(6,49) in PG(12,7)) 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]
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 (see above)
3. digital (23, 46, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 23, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(58, 81, 2385)-Net over F₇ — Digital

Digital (58, 81, 2385)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7⁸¹, 2385, F₇, 23) (dual of [2385, 2304, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(7⁸¹, 2402, F₇, 23) (dual of [2402, 2321, 24]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 2402Â | 7⁸âˆ’1, defining interval IÂ =Â [0,11], and minimum distance dÂ â‰¥ |{−11,−10,…,11}|+1Â =Â 24 (BCH-bound) [i]

(58, 81, 1145586)-Net in Base 7 — Upper bound on s

There is no (58, 81, 1145587)-net in base 7, because

1 times m-reduction [i] would yield (58, 80, 1145587)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 40 536418 519301 133958 856087 781014 737790 011690 789472 356625 022691 640959 > 7⁸⁰ [i]