MinT - Best Known (42, 42+19, s)-Nets in Base 7

Best Known (42, 42+19, s)-Nets in Base 7

(42, 42+19, 204)-Net over F₇ — Constructive and digital

Digital (42, 61, 204)-net over F₇, using

7¹ times duplication [i] based on digital (41, 60, 204)-net over F₇, using
- (u,Â u+v)-construction [i] based on
  1. 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]
  2. digital (21, 40, 102)-net over F₇, using
    - trace code for nets [i] based on digital (1, 20, 51)-net over F₄₉, using
      - net from sequence [i] based on digital (1, 50)-sequence over F₄₉ (see above)

(42, 42+19, 929)-Net over F₇ — Digital

Digital (42, 61, 929)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(42, 42+19, 297563)-Net in Base 7 — Upper bound on s

There is no (42, 61, 297564)-net in base 7, because

1 times m-reduction [i] would yield (42, 60, 297564)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 508 035379 264558 668282 248241 060188 268811 228020 472873 > 7⁶⁰ [i]