MinT - Best Known (4, 21, s)-Nets in Base 5

Best Known (4, 21, s)-Nets in Base 5

(4, 21, 18)-Net over F₅ — Constructive and digital

Digital (4, 21, 18)-net over F₅, using

net from sequence [i] based on digital (4, 17)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 4 and N(F) ≥ 18, using
  - an explicitly constructive algebraic function field [i]

(4, 21, 41)-Net over F₅ — Upper bound on s (digital)

There is no digital (4, 21, 42)-net over F₅, because

2 times m-reduction [i] would yield digital (4, 19, 42)-net over F₅, but
- extracting embedded orthogonal array [i] would yield linear OA(5¹⁹, 42, F₅, 15) (dual of [42, 23, 16]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(5¹⁸, 23, F₅, 15) (dual of [23, 5, 16]-code), but
      - residual code [i] would yield OA(5³, 7, S₅, 3), but
        bound for OAs with index unity [i]
    2. OA(5²³, 42, S₅, 19), but
      - discarding factors would yield OA(5²³, 41, S₅, 19), but
        the linear programming bound shows that MÂ â‰¥ 1431 381081 044673 919677 734375 / 117616 346784 > 5²³ [i]

(4, 21, 46)-Net in Base 5 — Upper bound on s

There is no (4, 21, 47)-net in base 5, because

extracting embedded orthogonal array [i] would yield OA(5²¹, 47, S₅, 17), but
- the linear programming bound shows that MÂ â‰¥ 2 226091 376121 462535 858154 296875 / 4601 222665 373363 > 5²¹ [i]