MinT - Best Known (3, 19, s)-Nets in Base 5

Best Known (3, 19, s)-Nets in Base 5

(3, 19, 16)-Net over F₅ — Constructive and digital

Digital (3, 19, 16)-net over F₅, using

net from sequence [i] based on digital (3, 15)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 3 and N(F) ≥ 16, using
  - an explicitly constructive algebraic function field [i]

(3, 19, 22)-Net over F₅ — Upper bound on s (digital)

There is no digital (3, 19, 23)-net over F₅, because

1 times m-reduction [i] would yield digital (3, 18, 23)-net over F₅, but
- extracting embedded orthogonal array [i] would yield 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]

(3, 19, 30)-Net in Base 5 — Upper bound on s

There is no (3, 19, 31)-net in base 5, because

1 times m-reduction [i] would yield (3, 18, 31)-net in base 5, but
- extracting embedded OOA [i] would yield OOA(5¹⁸, 31, S₅, 2, 15), but
  - the linear programming bound for OOAs shows that MÂ â‰¥ 3 517198 136814 630536 328002 356852 932502 557478 924518 706687 927246 093750 / 890186 821974 662831 442996 193641 186220 027713 081363 462721 > 5¹⁸ [i]