Best Known (168−35, 168, s)-Nets in Base 4
(168−35, 168, 1056)-Net over F4 — Constructive and digital
Digital (133, 168, 1056)-net over F4, using
- trace code for nets [i] based on digital (7, 42, 264)-net over F256, using
- net from sequence [i] based on digital (7, 263)-sequence over F256, using
(168−35, 168, 4387)-Net over F4 — Digital
Digital (133, 168, 4387)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4168, 4387, F4, 35) (dual of [4387, 4219, 36]-code), using
- 274 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 5 times 0, 1, 9 times 0, 1, 17 times 0, 1, 28 times 0, 1, 44 times 0, 1, 66 times 0, 1, 94 times 0) [i] based on linear OA(4157, 4102, F4, 35) (dual of [4102, 3945, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(33) [i] based on
- 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(4151, 4096, F4, 34) (dual of [4096, 3945, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [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(34) ⊂ Ce(33) [i] based on
- 274 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 5 times 0, 1, 9 times 0, 1, 17 times 0, 1, 28 times 0, 1, 44 times 0, 1, 66 times 0, 1, 94 times 0) [i] based on linear OA(4157, 4102, F4, 35) (dual of [4102, 3945, 36]-code), using
(168−35, 168, 1964151)-Net in Base 4 — Upper bound on s
There is no (133, 168, 1964152)-net in base 4, because
- 1 times m-reduction [i] would yield (133, 167, 1964152)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 34996 190843 798871 959594 160795 701193 699141 132294 789724 718111 416687 288962 613540 955283 354111 356952 275455 > 4167 [i]