Best Known (166, 166+46, s)-Nets in Base 4
(166, 166+46, 1056)-Net over F4 — Constructive and digital
Digital (166, 212, 1056)-net over F4, using
- trace code for nets [i] based on digital (7, 53, 264)-net over F256, using
- net from sequence [i] based on digital (7, 263)-sequence over F256, using
(166, 166+46, 4187)-Net over F4 — Digital
Digital (166, 212, 4187)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4212, 4187, F4, 46) (dual of [4187, 3975, 47]-code), using
- 78 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 1, 4 times 0, 1, 10 times 0, 1, 20 times 0, 1, 37 times 0) [i] based on linear OA(4205, 4102, F4, 46) (dual of [4102, 3897, 47]-code), using
- construction X applied to Ce(45) ⊂ Ce(44) [i] based on
- linear OA(4205, 4096, F4, 46) (dual of [4096, 3891, 47]-code), using an extension Ce(45) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,45], and designed minimum distance d ≥ |I|+1 = 46 [i]
- 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]
- 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(45) ⊂ Ce(44) [i] based on
- 78 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 1, 4 times 0, 1, 10 times 0, 1, 20 times 0, 1, 37 times 0) [i] based on linear OA(4205, 4102, F4, 46) (dual of [4102, 3897, 47]-code), using
(166, 166+46, 1113653)-Net in Base 4 — Upper bound on s
There is no (166, 212, 1113654)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 43 322980 337272 778471 764776 268759 696792 344646 701415 894919 401883 001921 100754 504937 135370 524834 163746 757337 202986 387312 234901 348352 > 4212 [i]