Best Known (2, 2+8, s)-Nets in Base 4
(2, 2+8, 10)-Net over F4 — Constructive and digital
Digital (2, 10, 10)-net over F4, using
- net from sequence [i] based on digital (2, 9)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 2 and N(F) ≥ 10, using
- a shift-net [i]
(2, 2+8, 14)-Net over F4 — Upper bound on s (digital)
There is no digital (2, 10, 15)-net over F4, because
- extracting embedded orthogonal array [i] would yield linear OA(410, 15, F4, 8) (dual of [15, 5, 9]-code), but
- residual code [i] would yield OA(42, 6, S4, 2), but
- bound for OAs with strength k = 2 [i]
- the Rao or (dual) Hamming bound shows that M ≥ 19 > 42 [i]
- residual code [i] would yield OA(42, 6, S4, 2), but
(2, 2+8, 20)-Net in Base 4 — Upper bound on s
There is no (2, 10, 21)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 1 064701 > 410 [i]