Best Known (164, 209, s)-Nets in Base 4
(164, 209, 1056)-Net over F4 — Constructive and digital
Digital (164, 209, 1056)-net over F4, using
- 41 times duplication [i] based on digital (163, 208, 1056)-net over F4, using
- trace code for nets [i] based on digital (7, 52, 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, 52, 264)-net over F256, using
(164, 209, 4285)-Net over F4 — Digital
Digital (164, 209, 4285)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4209, 4285, F4, 45) (dual of [4285, 4076, 46]-code), using
- 179 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 5 times 0, 1, 9 times 0, 1, 17 times 0, 1, 29 times 0, 1, 45 times 0, 1, 64 times 0) [i] based on linear OA(4199, 4096, F4, 45) (dual of [4096, 3897, 46]-code), using
- an extension Ce(44) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,44], and designed minimum distance d ≥ |I|+1 = 45 [i]
- 179 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 5 times 0, 1, 9 times 0, 1, 17 times 0, 1, 29 times 0, 1, 45 times 0, 1, 64 times 0) [i] based on linear OA(4199, 4096, F4, 45) (dual of [4096, 3897, 46]-code), using
(164, 209, 1485696)-Net in Base 4 — Upper bound on s
There is no (164, 209, 1485697)-net in base 4, because
- 1 times m-reduction [i] would yield (164, 208, 1485697)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 169231 466073 416493 494198 096593 295494 881019 281567 349460 487515 595137 722658 703763 481954 825745 638069 703168 702161 923717 944586 605760 > 4208 [i]