Best Known (188−40, 188, s)-Nets in Base 4
(188−40, 188, 1056)-Net over F4 — Constructive and digital
Digital (148, 188, 1056)-net over F4, using
- trace code for nets [i] based on digital (7, 47, 264)-net over F256, using
- net from sequence [i] based on digital (7, 263)-sequence over F256, using
(188−40, 188, 4232)-Net over F4 — Digital
Digital (148, 188, 4232)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4188, 4232, F4, 40) (dual of [4232, 4044, 41]-code), using
- 117 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 6 times 0, 1, 16 times 0, 1, 32 times 0, 1, 56 times 0) [i] based on linear OA(4182, 4109, F4, 40) (dual of [4109, 3927, 41]-code), using
- construction X applied to Ce(40) ⊂ Ce(37) [i] based on
- linear OA(4181, 4096, F4, 41) (dual of [4096, 3915, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(4169, 4096, F4, 38) (dual of [4096, 3927, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(41, 13, F4, 1) (dual of [13, 12, 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(40) ⊂ Ce(37) [i] based on
- 117 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 6 times 0, 1, 16 times 0, 1, 32 times 0, 1, 56 times 0) [i] based on linear OA(4182, 4109, F4, 40) (dual of [4109, 3927, 41]-code), using
(188−40, 188, 1263407)-Net in Base 4 — Upper bound on s
There is no (148, 188, 1263408)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 153916 107862 719963 620386 053394 824616 026795 482531 685409 256405 441455 227064 757466 255512 988755 142688 614626 196793 687474 > 4188 [i]