Best Known (168, 204, s)-Nets in Base 4
(168, 204, 1539)-Net over F4 — Constructive and digital
Digital (168, 204, 1539)-net over F4, using
- 6 times m-reduction [i] based on digital (168, 210, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 70, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- trace code for nets [i] based on digital (28, 70, 513)-net over F64, using
(168, 204, 16441)-Net over F4 — Digital
Digital (168, 204, 16441)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4204, 16441, F4, 36) (dual of [16441, 16237, 37]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4203, 16439, F4, 36) (dual of [16439, 16236, 37]-code), using
- construction X applied to Ce(36) ⊂ Ce(28) [i] based on
- linear OA(4190, 16384, F4, 37) (dual of [16384, 16194, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [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(413, 55, F4, 6) (dual of [55, 42, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- construction X applied to Ce(36) ⊂ Ce(28) [i] based on
- linear OA(4203, 16440, F4, 35) (dual of [16440, 16237, 36]-code), using Gilbert–Varšamov bound and bm = 4203 > Vbs−1(k−1) = 11 945125 857580 729260 179878 865504 250703 315515 051851 351294 258491 575614 048943 274549 646759 042413 816508 924402 194026 036133 570333 [i]
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(4203, 16439, F4, 36) (dual of [16439, 16236, 37]-code), using
- construction X with Varšamov bound [i] based on
(168, 204, large)-Net in Base 4 — Upper bound on s
There is no (168, 204, large)-net in base 4, because
- 34 times m-reduction [i] would yield (168, 170, large)-net in base 4, but