Best Known (163−12, 163, s)-Nets in Base 3
(163−12, 163, 2796200)-Net over F3 — Constructive and digital
Digital (151, 163, 2796200)-net over F3, using
- 31 times duplication [i] based on digital (150, 162, 2796200)-net over F3, using
- net defined by OOA [i] based on linear OOA(3162, 2796200, F3, 14, 12) (dual of [(2796200, 14), 39146638, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(3162, 8388601, F3, 2, 12) (dual of [(8388601, 2), 16777040, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3162, 8388602, F3, 2, 12) (dual of [(8388602, 2), 16777042, 13]-NRT-code), using
- trace code [i] based on linear OOA(981, 4194301, F9, 2, 12) (dual of [(4194301, 2), 8388521, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(981, 8388602, F9, 12) (dual of [8388602, 8388521, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(981, large, F9, 12) (dual of [large, large−81, 13]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 98−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(981, large, F9, 12) (dual of [large, large−81, 13]-code), using
- OOA 2-folding [i] based on linear OA(981, 8388602, F9, 12) (dual of [8388602, 8388521, 13]-code), using
- trace code [i] based on linear OOA(981, 4194301, F9, 2, 12) (dual of [(4194301, 2), 8388521, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3162, 8388602, F3, 2, 12) (dual of [(8388602, 2), 16777042, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(3162, 8388601, F3, 2, 12) (dual of [(8388601, 2), 16777040, 13]-NRT-code), using
- net defined by OOA [i] based on linear OOA(3162, 2796200, F3, 14, 12) (dual of [(2796200, 14), 39146638, 13]-NRT-code), using
(163−12, 163, large)-Net over F3 — Digital
Digital (151, 163, large)-net over F3, using
- 2 times m-reduction [i] based on digital (151, 165, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3165, large, F3, 14) (dual of [large, large−165, 15]-code), using
- 29 times code embedding in larger space [i] based on linear OA(3136, large, F3, 14) (dual of [large, large−136, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- 29 times code embedding in larger space [i] based on linear OA(3136, large, F3, 14) (dual of [large, large−136, 15]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3165, large, F3, 14) (dual of [large, large−165, 15]-code), using
(163−12, 163, large)-Net in Base 3 — Upper bound on s
There is no (151, 163, large)-net in base 3, because
- 10 times m-reduction [i] would yield (151, 153, large)-net in base 3, but