Best Known (38, 38+18, s)-Nets in Base 4
(38, 38+18, 195)-Net over F4 — Constructive and digital
Digital (38, 56, 195)-net over F4, using
- 1 times m-reduction [i] based on digital (38, 57, 195)-net over F4, using
- trace code for nets [i] based on digital (0, 19, 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, 19, 65)-net over F64, using
(38, 38+18, 255)-Net over F4 — Digital
Digital (38, 56, 255)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(456, 255, F4, 18) (dual of [255, 199, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(456, 272, F4, 18) (dual of [272, 216, 19]-code), using
- construction XX applied to Ce(17) ⊂ Ce(13) ⊂ Ce(12) [i] based on
- linear OA(451, 256, F4, 18) (dual of [256, 205, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(441, 256, F4, 14) (dual of [256, 215, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(437, 256, F4, 13) (dual of [256, 219, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(44, 15, F4, 3) (dual of [15, 11, 4]-code or 15-cap in PG(3,4)), using
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(17) ⊂ Ce(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(456, 272, F4, 18) (dual of [272, 216, 19]-code), using
(38, 38+18, 7698)-Net in Base 4 — Upper bound on s
There is no (38, 56, 7699)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 5197 422589 609300 345848 329163 007804 > 456 [i]