Best Known (168, 168+37, s)-Nets in Base 4
(168, 168+37, 1539)-Net over F4 — Constructive and digital
Digital (168, 205, 1539)-net over F4, using
- 5 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, 168+37, 14944)-Net over F4 — Digital
Digital (168, 205, 14944)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4205, 14944, F4, 37) (dual of [14944, 14739, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(4205, 16434, F4, 37) (dual of [16434, 16229, 38]-code), using
- 2 times code embedding in larger space [i] based on linear OA(4203, 16432, F4, 37) (dual of [16432, 16229, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(29) [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(4155, 16384, F4, 30) (dual of [16384, 16229, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(413, 48, F4, 6) (dual of [48, 35, 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(29) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(4203, 16432, F4, 37) (dual of [16432, 16229, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(4205, 16434, F4, 37) (dual of [16434, 16229, 38]-code), using
(168, 168+37, large)-Net in Base 4 — Upper bound on s
There is no (168, 205, large)-net in base 4, because
- 35 times m-reduction [i] would yield (168, 170, large)-net in base 4, but