MinT - Best Known (85−27, 85, s)-Nets in Base 7

Best Known (85−27, 85, s)-Nets in Base 7

(85−27, 85, 204)-Net over F₇ — Constructive and digital

Digital (58, 85, 204)-net over F₇, using

7¹ times duplication [i] based on digital (57, 84, 204)-net over F₇, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 28, 102)-net over F₇, using
    - trace code for nets [i] based on digital (1, 14, 51)-net over F₄₉, using
      - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
        Niederreiter sequence [i]
  2. digital (29, 56, 102)-net over F₇, using
    - trace code for nets [i] based on digital (1, 28, 51)-net over F₄₉, using
      - net from sequence [i] based on digital (1, 50)-sequence over F₄₉ (see above)

(85−27, 85, 1031)-Net over F₇ — Digital

Digital (58, 85, 1031)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(85−27, 85, 272819)-Net in Base 7 — Upper bound on s

There is no (58, 85, 272820)-net in base 7, because

1 times m-reduction [i] would yield (58, 84, 272820)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 97329 747740 325660 906249 819163 507210 205742 095831 734444 343874 794988 535129 > 7⁸⁴ [i]