Best Known (52, 52+18, s)-Nets in Base 4
(52, 52+18, 514)-Net over F4 — Constructive and digital
Digital (52, 70, 514)-net over F4, using
- trace code for nets [i] based on digital (17, 35, 257)-net over F16, using
- base reduction for projective spaces (embedding PG(17,256) in PG(34,16)) for nets [i] based on digital (0, 18, 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(17,256) in PG(34,16)) for nets [i] based on digital (0, 18, 257)-net over F256, using
(52, 52+18, 883)-Net over F4 — Digital
Digital (52, 70, 883)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(470, 883, F4, 18) (dual of [883, 813, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(470, 1023, F4, 18) (dual of [1023, 953, 19]-code), using
- the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- discarding factors / shortening the dual code based on linear OA(470, 1023, F4, 18) (dual of [1023, 953, 19]-code), using
(52, 52+18, 66569)-Net in Base 4 — Upper bound on s
There is no (52, 70, 66570)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 1 393946 923664 386366 475604 988760 461603 423783 > 470 [i]