Best Known (53, 53+18, s)-Nets in Base 4
(53, 53+18, 514)-Net over F4 — Constructive and digital
Digital (53, 71, 514)-net over F4, using
- base reduction for projective spaces (embedding PG(35,16) in PG(70,4)) for nets [i] based on digital (18, 36, 514)-net over F16, using
- trace code 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
- trace code for nets [i] based on digital (0, 18, 257)-net over F256, using
(53, 53+18, 964)-Net over F4 — Digital
Digital (53, 71, 964)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(471, 964, F4, 18) (dual of [964, 893, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(471, 1044, F4, 18) (dual of [1044, 973, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- linear OA(466, 1024, F4, 18) (dual of [1024, 958, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(451, 1024, F4, 14) (dual of [1024, 973, 15]-code), using an extension Ce(13) of 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]
- linear OA(45, 20, F4, 3) (dual of [20, 15, 4]-code or 20-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(471, 1044, F4, 18) (dual of [1044, 973, 19]-code), using
(53, 53+18, 77656)-Net in Base 4 — Upper bound on s
There is no (53, 71, 77657)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 5 575785 253494 547948 872303 150078 631868 283148 > 471 [i]