Best Known (220−23, 220, s)-Nets in Base 4
(220−23, 220, 762615)-Net over F4 — Constructive and digital
Digital (197, 220, 762615)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (4, 15, 15)-net over F4, using
- net from sequence [i] based on digital (4, 14)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 4 and N(F) ≥ 15, using
- net from sequence [i] based on digital (4, 14)-sequence 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 (4, 15, 15)-net over F4, using
(220−23, 220, 5495353)-Net over F4 — Digital
Digital (197, 220, 5495353)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4220, 5495353, F4, 23) (dual of [5495353, 5495133, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4220, large, F4, 23) (dual of [large, large−220, 24]-code), using
- 15 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]
- 15 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(4220, large, F4, 23) (dual of [large, large−220, 24]-code), using
(220−23, 220, large)-Net in Base 4 — Upper bound on s
There is no (197, 220, large)-net in base 4, because
- 21 times m-reduction [i] would yield (197, 199, large)-net in base 4, but