Best Known (43, 64, s)-Nets in Base 4
(43, 64, 195)-Net over F4 — Constructive and digital
Digital (43, 64, 195)-net over F4, using
- 41 times duplication [i] based on digital (42, 63, 195)-net over F4, using
- trace code for nets [i] based on digital (0, 21, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- trace code for nets [i] based on digital (0, 21, 65)-net over F64, using
(43, 64, 249)-Net over F4 — Digital
Digital (43, 64, 249)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(464, 249, F4, 21) (dual of [249, 185, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(464, 271, F4, 21) (dual of [271, 207, 22]-code), using
- 1 times code embedding in larger space [i] based on linear OA(463, 270, F4, 21) (dual of [270, 207, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(16) [i] based on
- linear OA(459, 256, F4, 21) (dual of [256, 197, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(449, 256, F4, 17) (dual of [256, 207, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(44, 14, F4, 3) (dual of [14, 10, 4]-code or 14-cap in PG(3,4)), using
- construction X applied to Ce(20) ⊂ Ce(16) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(463, 270, F4, 21) (dual of [270, 207, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(464, 271, F4, 21) (dual of [271, 207, 22]-code), using
(43, 64, 9364)-Net in Base 4 — Upper bound on s
There is no (43, 64, 9365)-net in base 4, because
- 1 times m-reduction [i] would yield (43, 63, 9365)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 85 144630 934099 447758 931092 979771 052628 > 463 [i]