Best Known (191−34, 191, s)-Nets in Base 4
(191−34, 191, 1539)-Net over F4 — Constructive and digital
Digital (157, 191, 1539)-net over F4, using
- t-expansion [i] based on digital (156, 191, 1539)-net over F4, using
- 1 times m-reduction [i] based on digital (156, 192, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 64, 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, 64, 513)-net over F64, using
- 1 times m-reduction [i] based on digital (156, 192, 1539)-net over F4, using
(191−34, 191, 15988)-Net over F4 — Digital
Digital (157, 191, 15988)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4191, 15988, F4, 34) (dual of [15988, 15797, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(4191, 16434, F4, 34) (dual of [16434, 16243, 35]-code), using
- 2 times code embedding in larger space [i] based on linear OA(4189, 16432, F4, 34) (dual of [16432, 16243, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(26) [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(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(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(33) ⊂ Ce(26) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(4189, 16432, F4, 34) (dual of [16432, 16243, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(4191, 16434, F4, 34) (dual of [16434, 16243, 35]-code), using
(191−34, 191, large)-Net in Base 4 — Upper bound on s
There is no (157, 191, large)-net in base 4, because
- 32 times m-reduction [i] would yield (157, 159, large)-net in base 4, but