Best Known (161−18, 161, s)-Nets in Base 4
(161−18, 161, 932067)-Net over F4 — Constructive and digital
Digital (143, 161, 932067)-net over F4, using
- 44 times duplication [i] based on digital (139, 157, 932067)-net over F4, using
- net defined by OOA [i] based on linear OOA(4157, 932067, F4, 18, 18) (dual of [(932067, 18), 16777049, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(4157, large, F4, 18) (dual of [large, large−157, 19]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- OA 9-folding and stacking [i] based on linear OA(4157, large, F4, 18) (dual of [large, large−157, 19]-code), using
- net defined by OOA [i] based on linear OOA(4157, 932067, F4, 18, 18) (dual of [(932067, 18), 16777049, 19]-NRT-code), using
(161−18, 161, 4194303)-Net over F4 — Digital
Digital (143, 161, 4194303)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4161, 4194303, F4, 2, 18) (dual of [(4194303, 2), 8388445, 19]-NRT-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(4157, 4194301, F4, 2, 18) (dual of [(4194301, 2), 8388445, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4157, 8388602, F4, 18) (dual of [8388602, 8388445, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(4157, large, F4, 18) (dual of [large, large−157, 19]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- discarding factors / shortening the dual code based on linear OA(4157, large, F4, 18) (dual of [large, large−157, 19]-code), using
- OOA 2-folding [i] based on linear OA(4157, 8388602, F4, 18) (dual of [8388602, 8388445, 19]-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(4157, 4194301, F4, 2, 18) (dual of [(4194301, 2), 8388445, 19]-NRT-code), using
(161−18, 161, large)-Net in Base 4 — Upper bound on s
There is no (143, 161, large)-net in base 4, because
- 16 times m-reduction [i] would yield (143, 145, large)-net in base 4, but