Best Known (82, 82+21, s)-Nets in Base 4
(82, 82+21, 1051)-Net over F4 — Constructive and digital
Digital (82, 103, 1051)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (9, 19, 23)-net over F4, using
- 4 times m-reduction [i] based on digital (9, 23, 23)-net over F4, using
- digital (63, 84, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 21, 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, 21, 257)-net over F256, using
- digital (9, 19, 23)-net over F4, using
(82, 82+21, 4242)-Net over F4 — Digital
Digital (82, 103, 4242)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4103, 4242, F4, 21) (dual of [4242, 4139, 22]-code), using
- 134 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 12 times 0, 1, 19 times 0, 1, 32 times 0, 1, 49 times 0) [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]
- 134 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 12 times 0, 1, 19 times 0, 1, 32 times 0, 1, 49 times 0) [i] based on linear OA(491, 4096, F4, 21) (dual of [4096, 4005, 22]-code), using
(82, 82+21, 2088648)-Net in Base 4 — Upper bound on s
There is no (82, 103, 2088649)-net in base 4, because
- 1 times m-reduction [i] would yield (82, 102, 2088649)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 25 711086 303975 105719 196380 230085 023572 301619 940397 267720 794736 > 4102 [i]