MinT - Best Known (1, 1+3, s)-Nets in Base 4

Best Known (1, 1+3, s)-Nets in Base 4

(1, 1+3, 17)-Net over F₄ — Constructive and digital

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]

(1, 1+3, 17)-Net over F₄ — Upper bound on s (digital)

There is no digital (1, 4, 18)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4⁴, 18, F₄, 3) (dual of [18, 14, 4]-code or 18-cap in PG(3,4)), but
- ovoid cap is optimal [i]

(1, 1+3, 21)-Net in Base 4 — Upper bound on s

There is no (1, 4, 22)-net in base 4, because

1 times m-reduction [i] would yield (1, 3, 22)-net in base 4, but
- mutually orthogonal hypercube bound [i]
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 67 > 4³ [i]