Best Known (168, 168+20, s)-Nets in Base 4
(168, 168+20, 838860)-Net over F4 — Constructive and digital
Digital (168, 188, 838860)-net over F4, using
- 47 times duplication [i] based on digital (161, 181, 838860)-net over F4, using
- t-expansion [i] based on digital (160, 181, 838860)-net over F4, using
- net defined by OOA [i] based on linear OOA(4181, 838860, F4, 21, 21) (dual of [(838860, 21), 17615879, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4181, 8388601, F4, 21) (dual of [8388601, 8388420, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4181, large, F4, 21) (dual of [large, large−181, 22]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- discarding factors / shortening the dual code based on linear OA(4181, large, F4, 21) (dual of [large, large−181, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4181, 8388601, F4, 21) (dual of [8388601, 8388420, 22]-code), using
- net defined by OOA [i] based on linear OOA(4181, 838860, F4, 21, 21) (dual of [(838860, 21), 17615879, 22]-NRT-code), using
- t-expansion [i] based on digital (160, 181, 838860)-net over F4, using
(168, 168+20, 4526284)-Net over F4 — Digital
Digital (168, 188, 4526284)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4188, 4526284, F4, 20) (dual of [4526284, 4526096, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(4188, large, F4, 20) (dual of [large, large−188, 21]-code), using
- 8 times code embedding in larger space [i] based on linear OA(4180, large, F4, 20) (dual of [large, large−180, 21]-code), using
- the primitive narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- 8 times code embedding in larger space [i] based on linear OA(4180, large, F4, 20) (dual of [large, large−180, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(4188, large, F4, 20) (dual of [large, large−188, 21]-code), using
(168, 168+20, large)-Net in Base 4 — Upper bound on s
There is no (168, 188, large)-net in base 4, because
- 18 times m-reduction [i] would yield (168, 170, large)-net in base 4, but