Best Known (212−23, 212, s)-Nets in Base 4
(212−23, 212, 762600)-Net over F4 — Constructive and digital
Digital (189, 212, 762600)-net over F4, using
- 47 times duplication [i] based on 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
(212−23, 212, 4194304)-Net over F4 — Digital
Digital (189, 212, 4194304)-net over F4, using
- 41 times duplication [i] based on digital (188, 211, 4194304)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4211, 4194304, F4, 2, 23) (dual of [(4194304, 2), 8388397, 24]-NRT-code), using
- 3 times NRT-code embedding in larger space [i] based on linear OOA(4205, 4194301, F4, 2, 23) (dual of [(4194301, 2), 8388397, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4205, 8388602, F4, 23) (dual of [8388602, 8388397, 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 2-folding [i] based on linear OA(4205, 8388602, F4, 23) (dual of [8388602, 8388397, 24]-code), using
- 3 times NRT-code embedding in larger space [i] based on linear OOA(4205, 4194301, F4, 2, 23) (dual of [(4194301, 2), 8388397, 24]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4211, 4194304, F4, 2, 23) (dual of [(4194304, 2), 8388397, 24]-NRT-code), using
(212−23, 212, large)-Net in Base 4 — Upper bound on s
There is no (189, 212, large)-net in base 4, because
- 21 times m-reduction [i] would yield (189, 191, large)-net in base 4, but