Best Known (224−50, 224, s)-Nets in Base 4
(224−50, 224, 1052)-Net over F4 — Constructive and digital
Digital (174, 224, 1052)-net over F4, using
- trace code for nets [i] based on digital (6, 56, 263)-net over F256, using
- net from sequence [i] based on digital (6, 262)-sequence over F256, using
(224−50, 224, 3877)-Net over F4 — Digital
Digital (174, 224, 3877)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4224, 3877, F4, 50) (dual of [3877, 3653, 51]-code), using
- discarding factors / shortening the dual code based on linear OA(4224, 4103, F4, 50) (dual of [4103, 3879, 51]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4223, 4102, F4, 50) (dual of [4102, 3879, 51]-code), using
- construction X applied to Ce(49) ⊂ Ce(48) [i] based on
- linear OA(4223, 4096, F4, 50) (dual of [4096, 3873, 51]-code), using an extension Ce(49) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,49], and designed minimum distance d ≥ |I|+1 = 50 [i]
- linear OA(4217, 4096, F4, 49) (dual of [4096, 3879, 50]-code), using an extension Ce(48) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,48], and designed minimum distance d ≥ |I|+1 = 49 [i]
- linear OA(40, 6, F4, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(49) ⊂ Ce(48) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4223, 4102, F4, 50) (dual of [4102, 3879, 51]-code), using
- discarding factors / shortening the dual code based on linear OA(4224, 4103, F4, 50) (dual of [4103, 3879, 51]-code), using
(224−50, 224, 841301)-Net in Base 4 — Upper bound on s
There is no (174, 224, 841302)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 726 852230 797814 428456 182015 357375 821913 402888 942616 951111 898074 463805 796555 018721 028487 620005 173113 183480 095140 900034 298903 583903 122760 > 4224 [i]