Best Known (168, 214, s)-Nets in Base 4
(168, 214, 1056)-Net over F4 — Constructive and digital
Digital (168, 214, 1056)-net over F4, using
- 42 times duplication [i] based on 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
- trace code for nets [i] based on digital (7, 53, 264)-net over F256, using
(168, 214, 4341)-Net over F4 — Digital
Digital (168, 214, 4341)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4214, 4341, F4, 46) (dual of [4341, 4127, 47]-code), using
- 230 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, 1, 61 times 0, 1, 89 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
- 230 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, 1, 61 times 0, 1, 89 times 0) [i] based on linear OA(4205, 4102, F4, 46) (dual of [4102, 3897, 47]-code), using
(168, 214, 1256331)-Net in Base 4 — Upper bound on s
There is no (168, 214, 1256332)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 693 177637 922911 153258 977951 243416 478261 783973 515594 783505 507962 923051 277759 954641 317861 608833 886639 919036 033607 442585 550383 889072 > 4214 [i]