Best Known (209−44, 209, s)-Nets in Base 4
(209−44, 209, 1060)-Net over F4 — Constructive and digital
Digital (165, 209, 1060)-net over F4, using
- 41 times duplication [i] based on digital (164, 208, 1060)-net over F4, using
- trace code for nets [i] based on digital (8, 52, 265)-net over F256, using
- net from sequence [i] based on digital (8, 264)-sequence over F256, using
- trace code for nets [i] based on digital (8, 52, 265)-net over F256, using
(209−44, 209, 4772)-Net over F4 — Digital
Digital (165, 209, 4772)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4209, 4772, F4, 44) (dual of [4772, 4563, 45]-code), using
- 666 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0, 1, 12 times 0, 1, 23 times 0, 1, 43 times 0, 1, 69 times 0, 1, 98 times 0, 1, 122 times 0, 1, 137 times 0, 1, 146 times 0) [i] based on linear OA(4198, 4095, F4, 44) (dual of [4095, 3897, 45]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(4199, 4096, F4, 45) (dual of [4096, 3897, 46]-code), using
- 666 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0, 1, 12 times 0, 1, 23 times 0, 1, 43 times 0, 1, 69 times 0, 1, 98 times 0, 1, 122 times 0, 1, 137 times 0, 1, 146 times 0) [i] based on linear OA(4198, 4095, F4, 44) (dual of [4095, 3897, 45]-code), using
(209−44, 209, 1582329)-Net in Base 4 — Upper bound on s
There is no (165, 209, 1582330)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 676930 463302 244140 949843 169057 695491 255671 147206 583725 326080 495484 767277 246026 959923 028536 874540 805085 198589 529059 450011 252756 > 4209 [i]