MinT - Best Known (4−3, 4, s)-Nets in Base 16

Best Known (4−3, 4, s)-Nets in Base 16

(4−3, 4, 257)-Net over F₁₆ — Constructive and digital

Digital (1, 4, 257)-net over F₁₆, using

net defined by OOA [i] based on linear OOA(16⁴, 257, F₁₆, 3, 3) (dual of [(257, 3), 767, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(16⁴, 257, F₁₆, 2, 3) (dual of [(257, 2), 510, 4]-NRT-code), using
  - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(16⁴, 257, F₁₆, 3) (dual of [257, 253, 4]-code or 257-cap in PG(3,16)), using
    - ovoid in PG(3,Â 16) [i]

(4−3, 4, 257)-Net over F₁₆ — Upper bound on s (digital)

There is no digital (1, 4, 258)-net over F₁₆, because

extracting embedded orthogonal array [i] would yield linear OA(16⁴, 258, F₁₆, 3) (dual of [258, 254, 4]-code or 258-cap in PG(3,16)), but
- ovoid cap is optimal [i]

(4−3, 4, 273)-Net in Base 16 — Upper bound on s

There is no (1, 4, 274)-net in base 16, because

1 times m-reduction [i] would yield (1, 3, 274)-net in base 16, but
- mutually orthogonal hypercube bound [i]
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 4111 > 16³ [i]