Best Known (82, 82+20, s)-Nets in Base 4
(82, 82+20, 1076)-Net over F4 — Constructive and digital
Digital (82, 102, 1076)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (12, 22, 48)-net over F4, using
- trace code for nets [i] based on digital (1, 11, 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, 11, 24)-net over F16, using
- digital (60, 80, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 20, 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, 20, 257)-net over F256, using
- digital (12, 22, 48)-net over F4, using
(82, 82+20, 4693)-Net over F4 — Digital
Digital (82, 102, 4693)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4102, 4693, F4, 20) (dual of [4693, 4591, 21]-code), using
- 586 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 5 times 0, 1, 11 times 0, 1, 22 times 0, 1, 37 times 0, 1, 62 times 0, 1, 97 times 0, 1, 142 times 0, 1, 198 times 0) [i] based on linear OA(490, 4095, F4, 20) (dual of [4095, 4005, 21]-code), using
- 1 times truncation [i] based on linear OA(491, 4096, F4, 21) (dual of [4096, 4005, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- 1 times truncation [i] based on linear OA(491, 4096, F4, 21) (dual of [4096, 4005, 22]-code), using
- 586 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 5 times 0, 1, 11 times 0, 1, 22 times 0, 1, 37 times 0, 1, 62 times 0, 1, 97 times 0, 1, 142 times 0, 1, 198 times 0) [i] based on linear OA(490, 4095, F4, 20) (dual of [4095, 4005, 21]-code), using
(82, 82+20, 2088648)-Net in Base 4 — Upper bound on s
There is no (82, 102, 2088649)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 25 711086 303975 105719 196380 230085 023572 301619 940397 267720 794736 > 4102 [i]