Best Known (37, 50, s)-Nets in Base 4
(37, 50, 514)-Net over F4 — Constructive and digital
Digital (37, 50, 514)-net over F4, using
- trace code for nets [i] based on digital (12, 25, 257)-net over F16, using
- base reduction for projective spaces (embedding PG(12,256) in PG(24,16)) for nets [i] based on digital (0, 13, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- base reduction for projective spaces (embedding PG(12,256) in PG(24,16)) for nets [i] based on digital (0, 13, 257)-net over F256, using
(37, 50, 779)-Net over F4 — Digital
Digital (37, 50, 779)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(450, 779, F4, 13) (dual of [779, 729, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(450, 1023, F4, 13) (dual of [1023, 973, 14]-code), using
- the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- discarding factors / shortening the dual code based on linear OA(450, 1023, F4, 13) (dual of [1023, 973, 14]-code), using
(37, 50, 82394)-Net in Base 4 — Upper bound on s
There is no (37, 50, 82395)-net in base 4, because
- 1 times m-reduction [i] would yield (37, 49, 82395)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 316914 775106 740761 253432 240918 > 449 [i]