Best Known (158, 158+34, s)-Nets in Base 4
(158, 158+34, 1539)-Net over F4 — Constructive and digital
Digital (158, 192, 1539)-net over F4, using
- 3 times m-reduction [i] based on digital (158, 195, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 65, 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, 65, 513)-net over F64, using
(158, 158+34, 16443)-Net over F4 — Digital
Digital (158, 192, 16443)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4192, 16443, F4, 34) (dual of [16443, 16251, 35]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4190, 16440, F4, 34) (dual of [16440, 16250, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(25) [i] based on
- linear OA(4176, 16384, F4, 34) (dual of [16384, 16208, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(4134, 16384, F4, 26) (dual of [16384, 16250, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(414, 56, F4, 7) (dual of [56, 42, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(414, 65, F4, 7) (dual of [65, 51, 8]-code), using
- a “GraXX†code from Grassl’s database [i]
- discarding factors / shortening the dual code based on linear OA(414, 65, F4, 7) (dual of [65, 51, 8]-code), using
- construction X applied to Ce(33) ⊂ Ce(25) [i] based on
- linear OA(4190, 16441, F4, 32) (dual of [16441, 16251, 33]-code), using Gilbert–Varšamov bound and bm = 4190 > Vbs−1(k−1) = 3603 092496 508275 938237 839199 608426 398294 719346 727132 500788 600047 218404 713769 295256 698755 178995 345763 906858 872524 [i]
- linear OA(41, 2, F4, 1) (dual of [2, 1, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(4190, 16440, F4, 34) (dual of [16440, 16250, 35]-code), using
- construction X with Varšamov bound [i] based on
(158, 158+34, large)-Net in Base 4 — Upper bound on s
There is no (158, 192, large)-net in base 4, because
- 32 times m-reduction [i] would yield (158, 160, large)-net in base 4, but