Best Known (64, 85, s)-Nets in Base 4
(64, 85, 1028)-Net over F4 — Constructive and digital
Digital (64, 85, 1028)-net over F4, using
- 41 times duplication [i] based on 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
(64, 85, 1102)-Net over F4 — Digital
Digital (64, 85, 1102)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(485, 1102, F4, 21) (dual of [1102, 1017, 22]-code), using
- 69 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) [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]
- 69 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) [i] based on linear OA(476, 1024, F4, 21) (dual of [1024, 948, 22]-code), using
(64, 85, 172242)-Net in Base 4 — Upper bound on s
There is no (64, 85, 172243)-net in base 4, because
- 1 times m-reduction [i] would yield (64, 84, 172243)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 374 164713 053285 795715 908278 783041 889769 170437 349030 > 484 [i]