Best Known (145−30, 145, s)-Nets in Base 4
(145−30, 145, 1055)-Net over F4 — Constructive and digital
Digital (115, 145, 1055)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (10, 25, 27)-net over F4, using
- net from sequence [i] based on digital (10, 26)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 10 and N(F) ≥ 27, using
- net from sequence [i] based on digital (10, 26)-sequence over F4, using
- digital (90, 120, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 30, 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, 30, 257)-net over F256, using
- digital (10, 25, 27)-net over F4, using
(145−30, 145, 4288)-Net over F4 — Digital
Digital (115, 145, 4288)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4145, 4288, F4, 30) (dual of [4288, 4143, 31]-code), using
- 174 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 10 times 0, 1, 16 times 0, 1, 26 times 0, 1, 41 times 0, 1, 62 times 0) [i] based on linear OA(4133, 4102, F4, 30) (dual of [4102, 3969, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(28) [i] based on
- linear OA(4133, 4096, F4, 30) (dual of [4096, 3963, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- 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(40, 6, F4, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(29) ⊂ Ce(28) [i] based on
- 174 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 10 times 0, 1, 16 times 0, 1, 26 times 0, 1, 41 times 0, 1, 62 times 0) [i] based on linear OA(4133, 4102, F4, 30) (dual of [4102, 3969, 31]-code), using
(145−30, 145, 1414343)-Net in Base 4 — Upper bound on s
There is no (115, 145, 1414344)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 1989 299928 351977 934673 853313 628322 419061 334151 532276 102970 066397 557962 195736 738987 092872 > 4145 [i]