Best Known (128−27, 128, s)-Nets in Base 4
(128−27, 128, 1049)-Net over F4 — Constructive and digital
Digital (101, 128, 1049)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (7, 20, 21)-net over F4, using
- net from sequence [i] based on digital (7, 20)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 7 and N(F) ≥ 21, using
- net from sequence [i] based on digital (7, 20)-sequence over F4, using
- digital (81, 108, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 27, 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, 27, 257)-net over F256, using
- digital (7, 20, 21)-net over F4, using
(128−27, 128, 3862)-Net over F4 — Digital
Digital (101, 128, 3862)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4128, 3862, F4, 27) (dual of [3862, 3734, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4128, 4127, F4, 27) (dual of [4127, 3999, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(21) [i] based on
- linear OA(4121, 4096, F4, 27) (dual of [4096, 3975, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(497, 4096, F4, 22) (dual of [4096, 3999, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(47, 31, F4, 4) (dual of [31, 24, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(26) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(4128, 4127, F4, 27) (dual of [4127, 3999, 28]-code), using
(128−27, 128, 1438615)-Net in Base 4 — Upper bound on s
There is no (101, 128, 1438616)-net in base 4, because
- 1 times m-reduction [i] would yield (101, 127, 1438616)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 28948 100212 342868 945383 371791 894588 646521 765450 840747 248606 710555 093300 295630 > 4127 [i]