MinT - Best Known (72−23, 72, s)-Nets in Base 7

Best Known (72−23, 72, s)-Nets in Base 7

(72−23, 72, 204)-Net over F₇ — Constructive and digital

Digital (49, 72, 204)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (13, 24, 102)-net over F₇, using
  - trace code for nets [i] based on digital (1, 12, 51)-net over F₄₉, using
    - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
      - Niederreiter sequence [i]
2. digital (25, 48, 102)-net over F₇, using
  - trace code for nets [i] based on digital (1, 24, 51)-net over F₄₉, using
    - net from sequence [i] based on digital (1, 50)-sequence over F₄₉ (see above)

(72−23, 72, 891)-Net over F₇ — Digital

Digital (49, 72, 891)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(72−23, 72, 233116)-Net in Base 7 — Upper bound on s

There is no (49, 72, 233117)-net in base 7, because

1 times m-reduction [i] would yield (49, 71, 233117)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 1 004526 661226 431273 944390 994394 418311 300700 775602 568575 789367 > 7⁷¹ [i]