Best Known (143, 180, s)-Nets in Base 4
(143, 180, 1060)-Net over F4 — Constructive and digital
Digital (143, 180, 1060)-net over F4, using
- trace code for nets [i] based on digital (8, 45, 265)-net over F256, using
- net from sequence [i] based on digital (8, 264)-sequence over F256, using
(143, 180, 4900)-Net over F4 — Digital
Digital (143, 180, 4900)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4180, 4900, F4, 37) (dual of [4900, 4720, 38]-code), using
- 781 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 8 times 0, 1, 13 times 0, 1, 22 times 0, 1, 34 times 0, 1, 52 times 0, 1, 75 times 0, 1, 101 times 0, 1, 129 times 0, 1, 154 times 0, 1, 172 times 0) [i] based on linear OA(4164, 4103, F4, 37) (dual of [4103, 3939, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(34) [i] based on
- linear OA(4163, 4096, F4, 37) (dual of [4096, 3933, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(4157, 4096, F4, 35) (dual of [4096, 3939, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(41, 7, F4, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(36) ⊂ Ce(34) [i] based on
- 781 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 8 times 0, 1, 13 times 0, 1, 22 times 0, 1, 34 times 0, 1, 52 times 0, 1, 75 times 0, 1, 101 times 0, 1, 129 times 0, 1, 154 times 0, 1, 172 times 0) [i] based on linear OA(4164, 4103, F4, 37) (dual of [4103, 3939, 38]-code), using
(143, 180, 2444321)-Net in Base 4 — Upper bound on s
There is no (143, 180, 2444322)-net in base 4, because
- 1 times m-reduction [i] would yield (143, 179, 2444322)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 587136 983104 646113 726397 476904 789241 841275 290730 285751 127353 196627 432911 162231 780887 961087 831249 041844 190368 > 4179 [i]