Best Known (11, 19, s)-Nets in Base 4
(11, 19, 48)-Net over F4 — Constructive and digital
Digital (11, 19, 48)-net over F4, using
- 1 times m-reduction [i] based on digital (11, 20, 48)-net over F4, using
- trace code for nets [i] based on digital (1, 10, 24)-net over F16, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 1 and N(F) ≥ 24, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- trace code for nets [i] based on digital (1, 10, 24)-net over F16, using
(11, 19, 60)-Net over F4 — Digital
Digital (11, 19, 60)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(419, 60, F4, 8) (dual of [60, 41, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(419, 63, F4, 8) (dual of [63, 44, 9]-code), using
- the primitive cyclic code C(A) with length 63 = 43−1, defining set A = {0,1,2,3,5,31,47}, and minimum distance d ≥ |{−2,−1,…,5}|+1 = 9 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(419, 63, F4, 8) (dual of [63, 44, 9]-code), using
(11, 19, 531)-Net in Base 4 — Upper bound on s
There is no (11, 19, 532)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 276138 961204 > 419 [i]