MinT - Best Known (31−17, 31, s)-Nets in Base 7

Best Known (31−17, 31, s)-Nets in Base 7

(31−17, 31, 26)-Net over F₇ — Constructive and digital

Digital (14, 31, 26)-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 (5, 22, 13)-net over F₇, using
  - net from sequence [i] based on digital (5, 12)-sequence over F₇, using
    - Niederreiter sequence [i]

(31−17, 31, 48)-Net over F₇ — Digital

Digital (14, 31, 48)-net over F₇, using

t-expansion [i] based on digital (13, 31, 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]

(31−17, 31, 921)-Net in Base 7 — Upper bound on s

There is no (14, 31, 922)-net in base 7, because

1 times m-reduction [i] would yield (14, 30, 922)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 22 681410 696167 562241 386385 > 7³⁰ [i]