Best Known (189, 189+52, s)-Nets in Base 4
(189, 189+52, 1539)-Net over F4 — Constructive and digital
Digital (189, 241, 1539)-net over F4, using
- 41 times duplication [i] based on digital (188, 240, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 80, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- trace code for nets [i] based on digital (28, 80, 513)-net over F64, using
(189, 189+52, 4658)-Net over F4 — Digital
Digital (189, 241, 4658)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4241, 4658, F4, 52) (dual of [4658, 4417, 53]-code), using
- 556 step Varšamov–Edel lengthening with (ri) = (1, 9 times 0, 1, 29 times 0, 1, 65 times 0, 1, 95 times 0, 1, 111 times 0, 1, 118 times 0, 1, 122 times 0) [i] based on linear OA(4234, 4095, F4, 52) (dual of [4095, 3861, 53]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(4235, 4096, F4, 53) (dual of [4096, 3861, 54]-code), using
- 556 step Varšamov–Edel lengthening with (ri) = (1, 9 times 0, 1, 29 times 0, 1, 65 times 0, 1, 95 times 0, 1, 111 times 0, 1, 118 times 0, 1, 122 times 0) [i] based on linear OA(4234, 4095, F4, 52) (dual of [4095, 3861, 53]-code), using
(189, 189+52, 1339053)-Net in Base 4 — Upper bound on s
There is no (189, 241, 1339054)-net in base 4, because
- 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]