Best Known (66, 66+21, s)-Nets in Base 4
(66, 66+21, 1028)-Net over F4 — Constructive and digital
Digital (66, 87, 1028)-net over F4, using
- 1 times m-reduction [i] based on digital (66, 88, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 22, 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, 22, 257)-net over F256, using
(66, 66+21, 1194)-Net over F4 — Digital
Digital (66, 87, 1194)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(487, 1194, F4, 21) (dual of [1194, 1107, 22]-code), using
- 159 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 0, 1, 6 times 0, 1, 10 times 0, 1, 17 times 0, 1, 25 times 0, 1, 37 times 0, 1, 51 times 0) [i] based on linear OA(476, 1024, F4, 21) (dual of [1024, 948, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- 159 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 0, 1, 6 times 0, 1, 10 times 0, 1, 17 times 0, 1, 25 times 0, 1, 37 times 0, 1, 51 times 0) [i] based on linear OA(476, 1024, F4, 21) (dual of [1024, 948, 22]-code), using
(66, 66+21, 227277)-Net in Base 4 — Upper bound on s
There is no (66, 87, 227278)-net in base 4, because
- 1 times m-reduction [i] would yield (66, 86, 227278)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 5986 502708 922841 467052 259569 746217 640599 584216 233206 > 486 [i]