MinT - Best Known (29−16, 29, s)-Nets in Base 7

Best Known (29−16, 29, s)-Nets in Base 7

(29−16, 29, 24)-Net over F₇ — Constructive and digital

Digital (13, 29, 24)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (1, 9, 13)-net over F₇, using
  - 4 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
    - a shift-net [i]
2. digital (4, 20, 12)-net over F₇, using
  - net from sequence [i] based on digital (4, 11)-sequence over F₇, using
    - Niederreiter sequence [i]
  - Niederreiter sequence (Bratleyâ€“Foxâ€“Niederreiter implementation) with equidistant coordinate [i]

(29−16, 29, 48)-Net over F₇ — Digital

Digital (13, 29, 48)-net over F₇, using

net from sequence [i] based on digital (13, 47)-sequence over F₇, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₇ with g(F) = 13 and N(F) ≥ 48, using
  - an algebraic function field by Teo [i]

(29−16, 29, 721)-Net in Base 7 — Upper bound on s

There is no (13, 29, 722)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 3 244256 844616 422911 379985 > 7²⁹ [i]