Best Known (133−28, 133, s)-Nets in Base 4
(133−28, 133, 1049)-Net over F4 — Constructive and digital
Digital (105, 133, 1049)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (7, 21, 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 (84, 112, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 28, 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, 28, 257)-net over F256, using
- digital (7, 21, 21)-net over F4, using
(133−28, 133, 3986)-Net over F4 — Digital
Digital (105, 133, 3986)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4133, 3986, F4, 28) (dual of [3986, 3853, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(4133, 4121, F4, 28) (dual of [4121, 3988, 29]-code), using
- 1 times truncation [i] based on linear OA(4134, 4122, F4, 29) (dual of [4122, 3988, 30]-code), using
- construction XX applied to Ce(28) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- linear OA(4127, 4096, F4, 29) (dual of [4096, 3969, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4109, 4096, F4, 25) (dual of [4096, 3987, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(4103, 4096, F4, 23) (dual of [4096, 3993, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(45, 24, F4, 3) (dual of [24, 19, 4]-code or 24-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- linear OA(41, 2, F4, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(28) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- 1 times truncation [i] based on linear OA(4134, 4122, F4, 29) (dual of [4122, 3988, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4133, 4121, F4, 28) (dual of [4121, 3988, 29]-code), using
(133−28, 133, 1056578)-Net in Base 4 — Upper bound on s
There is no (105, 133, 1056579)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 118 571719 305314 642239 281208 554463 518975 056613 396735 353716 426574 762403 463418 166064 > 4133 [i]