Best Known (189, 189+53, s)-Nets in Base 4
(189, 189+53, 1056)-Net over F4 — Constructive and digital
Digital (189, 242, 1056)-net over F4, using
- 42 times duplication [i] based on digital (187, 240, 1056)-net over F4, using
- trace code for nets [i] based on digital (7, 60, 264)-net over F256, using
- net from sequence [i] based on digital (7, 263)-sequence over F256, using
- trace code for nets [i] based on digital (7, 60, 264)-net over F256, using
(189, 189+53, 4318)-Net over F4 — Digital
Digital (189, 242, 4318)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4242, 4318, F4, 53) (dual of [4318, 4076, 54]-code), using
- 215 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 8 times 0, 1, 19 times 0, 1, 37 times 0, 1, 60 times 0, 1, 83 times 0) [i] based on linear OA(4235, 4096, F4, 53) (dual of [4096, 3861, 54]-code), using
- an extension Ce(52) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,52], and designed minimum distance d ≥ |I|+1 = 53 [i]
- 215 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 8 times 0, 1, 19 times 0, 1, 37 times 0, 1, 60 times 0, 1, 83 times 0) [i] based on linear OA(4235, 4096, F4, 53) (dual of [4096, 3861, 54]-code), using
(189, 189+53, 1339053)-Net in Base 4 — Upper bound on s
There is no (189, 242, 1339054)-net in base 4, because
- 1 times m-reduction [i] would yield (189, 241, 1339054)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 12 487124 808904 637777 649906 583033 162817 234929 371587 916231 960591 050329 152690 145859 168315 601695 544008 879662 665258 608456 603860 762237 195867 537478 354620 > 4241 [i]