MinT - Best Known (30, 30+95, s)-Nets in Base 5

Best Known (30, 30+95, s)-Nets in Base 5

(30, 30+95, 51)-Net over F₅ — Constructive and digital

Digital (30, 125, 51)-net over F₅, using

t-expansion [i] based on digital (22, 125, 51)-net over F₅, using
- net from sequence [i] based on digital (22, 50)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 22 and N(F) ≥ 51, using
    - an explicitly constructive algebraic function field [i]

(30, 30+95, 58)-Net over F₅ — Digital

Digital (30, 125, 58)-net over F₅, using

net from sequence [i] based on digital (30, 57)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 30 and N(F) ≥ 58, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(30, 30+95, 284)-Net over F₅ — Upper bound on s (digital)

There is no digital (30, 125, 285)-net over F₅, because

extracting embedded orthogonal array [i] would yield linear OA(5¹²⁵, 285, F₅, 95) (dual of [285, 160, 96]-code), but
- residual code [i] would yield OA(5³⁰, 189, S₅, 19), but
  - 1 times truncation [i] would yield OA(5²⁹, 188, S₅, 18), but
    - the linear programming bound shows that MÂ â‰¥ 22 829644 377403 021434 618461 608886 718750 / 121561 995449 060039 > 5²⁹ [i]

(30, 30+95, 287)-Net in Base 5 — Upper bound on s

There is no (30, 125, 288)-net in base 5, because

1 times m-reduction [i] would yield (30, 124, 288)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 525 269864 298513 062797 657017 686573 129855 088481 129854 031767 631167 867953 753465 211397 457025 > 5¹²⁴ [i]