Best Known (28, 28+8, s)-Nets in Base 4
(28, 28+8, 1033)-Net over F4 — Constructive and digital
Digital (28, 36, 1033)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 5)-net over F4, using
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 0 and N(F) ≥ 5, using
- the rational function field F4(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- digital (24, 32, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 8, 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, 8, 257)-net over F256, using
- digital (0, 4, 5)-net over F4, using
(28, 28+8, 3240)-Net over F4 — Digital
Digital (28, 36, 3240)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(436, 3240, F4, 8) (dual of [3240, 3204, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(436, 4096, F4, 8) (dual of [4096, 4060, 9]-code), using
- 1 times truncation [i] based on linear OA(437, 4097, F4, 9) (dual of [4097, 4060, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(437, 4097, F4, 9) (dual of [4097, 4060, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(436, 4096, F4, 8) (dual of [4096, 4060, 9]-code), using
(28, 28+8, 193403)-Net in Base 4 — Upper bound on s
There is no (28, 36, 193404)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 4722 381121 003900 374934 > 436 [i]