Best Known (144, 144+33, s)-Nets in Base 4
(144, 144+33, 1158)-Net over F4 — Constructive and digital
Digital (144, 177, 1158)-net over F4, using
- 41 times duplication [i] based on digital (143, 176, 1158)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (28, 44, 130)-net over F4, using
- trace code for nets [i] based on digital (6, 22, 65)-net over F16, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- the Hermitian function field over F16 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- trace code for nets [i] based on digital (6, 22, 65)-net over F16, using
- digital (99, 132, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 33, 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, 33, 257)-net over F256, using
- digital (28, 44, 130)-net over F4, using
- (u, u+v)-construction [i] based on
(144, 144+33, 10816)-Net over F4 — Digital
Digital (144, 177, 10816)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4177, 10816, F4, 33) (dual of [10816, 10639, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(4177, 16415, F4, 33) (dual of [16415, 16238, 34]-code), using
- construction XX applied to Ce(32) ⊂ Ce(28) ⊂ Ce(26) [i] based on
- linear OA(4169, 16384, F4, 33) (dual of [16384, 16215, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(4148, 16384, F4, 29) (dual of [16384, 16236, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4141, 16384, F4, 27) (dual of [16384, 16243, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(45, 28, F4, 3) (dual of [28, 23, 4]-code or 28-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
- linear OA(41, 3, F4, 1) (dual of [3, 2, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- Reed–Solomon code RS(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- construction XX applied to Ce(32) ⊂ Ce(28) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(4177, 16415, F4, 33) (dual of [16415, 16238, 34]-code), using
(144, 144+33, large)-Net in Base 4 — Upper bound on s
There is no (144, 177, large)-net in base 4, because
- 31 times m-reduction [i] would yield (144, 146, large)-net in base 4, but