Best Known (78, 78+20, s)-Nets in Base 4
(78, 78+20, 1049)-Net over F4 — Constructive and digital
Digital (78, 98, 1049)-net over F4, using
- 41 times duplication [i] based on digital (77, 97, 1049)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (7, 17, 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 (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 (7, 17, 21)-net over F4, using
- (u, u+v)-construction [i] based on
(78, 78+20, 4186)-Net over F4 — Digital
Digital (78, 98, 4186)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(498, 4186, F4, 20) (dual of [4186, 4088, 21]-code), using
- 83 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) [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
- 83 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) [i] based on linear OA(490, 4095, F4, 20) (dual of [4095, 4005, 21]-code), using
(78, 78+20, 1199610)-Net in Base 4 — Upper bound on s
There is no (78, 98, 1199611)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 100434 348661 557532 470815 417745 160371 032886 967963 601725 730803 > 498 [i]