MinT - Best Known (19−7, 19, s)-Nets in Base 4

Best Known (19−7, 19, s)-Nets in Base 4

(19−7, 19, 71)-Net over F₄ — Constructive and digital

Digital (12, 19, 71)-net over F₄, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 1, 17)-net over F₄, using
  - s-reduction based on digital (0, 1, s)-net over F₄ with arbitrarily large s, using
    - construction for strength kÂ =Â 1 [i]
2. digital (1, 3, 17)-net over F₄, using
  - s-reduction based on digital (1, 3, 21)-net over F₄, using
    - kÂ =Â 2 construction [i]
3. digital (1, 4, 17)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4⁴, 17, F₄, 3, 3) (dual of [(17, 3), 47, 4]-NRT-code), using
    - appending kth column [i] based on linear OOA(4⁴, 17, F₄, 2, 3) (dual of [(17, 2), 30, 4]-NRT-code), using
      - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(4⁴, 17, F₄, 3) (dual of [17, 13, 4]-code or 17-cap in PG(3,4)), using
        ovoid in PG(3,Â 4) [i]
4. digital (4, 11, 20)-net over F₄, using
  - linear code embedding found in a computer search [i]

(19−7, 19, 125)-Net over F₄ — Digital

Digital (12, 19, 125)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁹, 125, F₄, 7) (dual of [125, 106, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁹, 172, F₄, 7) (dual of [172, 153, 8]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(4¹⁸, 171, F₄, 7) (dual of [171, 153, 8]-code), using
    - a â€œGraâ€ code from Grasslâ€™s database [i]

(19−7, 19, 2478)-Net in Base 4 — Upper bound on s

There is no (12, 19, 2479)-net in base 4, because

1 times m-reduction [i] would yield (12, 18, 2479)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 68721 472072 > 4¹⁸ [i]