Best Known (222−23, 222, s)-Nets in Base 4
(222−23, 222, 762618)-Net over F4 — Constructive and digital
Digital (199, 222, 762618)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (6, 17, 18)-net over F4, using
- 1 times m-reduction [i] based on digital (6, 18, 18)-net over F4, using
- digital (182, 205, 762600)-net over F4, using
- net defined by OOA [i] based on linear OOA(4205, 762600, F4, 23, 23) (dual of [(762600, 23), 17539595, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(4205, 8388601, F4, 23) (dual of [8388601, 8388396, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4205, large, F4, 23) (dual of [large, large−205, 24]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,22], and designed minimum distance d ≥ |I|+1 = 24 [i]
- discarding factors / shortening the dual code based on linear OA(4205, large, F4, 23) (dual of [large, large−205, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(4205, 8388601, F4, 23) (dual of [8388601, 8388396, 24]-code), using
- net defined by OOA [i] based on linear OOA(4205, 762600, F4, 23, 23) (dual of [(762600, 23), 17539595, 24]-NRT-code), using
- digital (6, 17, 18)-net over F4, using
(222−23, 222, 6270972)-Net over F4 — Digital
Digital (199, 222, 6270972)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4222, 6270972, F4, 23) (dual of [6270972, 6270750, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4222, large, F4, 23) (dual of [large, large−222, 24]-code), using
- 17 times code embedding in larger space [i] based on linear OA(4205, large, F4, 23) (dual of [large, large−205, 24]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,22], and designed minimum distance d ≥ |I|+1 = 24 [i]
- 17 times code embedding in larger space [i] based on linear OA(4205, large, F4, 23) (dual of [large, large−205, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4222, large, F4, 23) (dual of [large, large−222, 24]-code), using
(222−23, 222, large)-Net in Base 4 — Upper bound on s
There is no (199, 222, large)-net in base 4, because
- 21 times m-reduction [i] would yield (199, 201, large)-net in base 4, but