Best Known (146−29, 146, s)-Nets in Base 4
(146−29, 146, 1076)-Net over F4 — Constructive and digital
Digital (117, 146, 1076)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (16, 30, 48)-net over F4, using
- trace code for nets [i] based on digital (1, 15, 24)-net over F16, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 1 and N(F) ≥ 24, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- trace code for nets [i] based on digital (1, 15, 24)-net over F16, using
- digital (87, 116, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 29, 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, 29, 257)-net over F256, using
- digital (16, 30, 48)-net over F4, using
(146−29, 146, 5223)-Net over F4 — Digital
Digital (117, 146, 5223)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4146, 5223, F4, 29) (dual of [5223, 5077, 30]-code), using
- 1108 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 11 times 0, 1, 18 times 0, 1, 27 times 0, 1, 40 times 0, 1, 58 times 0, 1, 80 times 0, 1, 107 times 0, 1, 138 times 0, 1, 170 times 0, 1, 202 times 0, 1, 229 times 0) [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]
- 1108 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 11 times 0, 1, 18 times 0, 1, 27 times 0, 1, 40 times 0, 1, 58 times 0, 1, 80 times 0, 1, 107 times 0, 1, 138 times 0, 1, 170 times 0, 1, 202 times 0, 1, 229 times 0) [i] based on linear OA(4127, 4096, F4, 29) (dual of [4096, 3969, 30]-code), using
(146−29, 146, 3467020)-Net in Base 4 — Upper bound on s
There is no (117, 146, 3467021)-net in base 4, because
- 1 times m-reduction [i] would yield (117, 145, 3467021)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 1989 295993 772267 955268 384900 041044 956751 010965 076362 878901 810994 322098 424557 726035 166256 > 4145 [i]