MinT - Best Known (131−69, 131, s)-Nets in Base 5

Best Known (131−69, 131, s)-Nets in Base 5

(131−69, 131, 82)-Net over F₅ — Constructive and digital

Digital (62, 131, 82)-net over F₅, using

t-expansion [i] based on digital (48, 131, 82)-net over F₅, using
- net from sequence [i] based on digital (48, 81)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 48 and N(F) ≥ 82, using
    - an explicitly constructive algebraic function field [i]

(131−69, 131, 120)-Net over F₅ — Digital

Digital (62, 131, 120)-net over F₅, using

t-expansion [i] based on digital (61, 131, 120)-net over F₅, using
- net from sequence [i] based on digital (61, 119)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 61 and N(F) ≥ 120, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(131−69, 131, 1567)-Net in Base 5 — Upper bound on s

There is no (62, 131, 1568)-net in base 5, because

1 times m-reduction [i] would yield (62, 130, 1568)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 7 480338 952375 353137 037963 970926 970035 683046 003977 179252 641589 710562 718899 816343 639188 242945 > 5¹³⁰ [i]