Best Known (212, 212+23, s)-Nets in Base 3
(212, 212+23, 762600)-Net over F3 — Constructive and digital
Digital (212, 235, 762600)-net over F3, using
- 39 times duplication [i] based on digital (203, 226, 762600)-net over F3, using
- net defined by OOA [i] based on linear OOA(3226, 762600, F3, 23, 23) (dual of [(762600, 23), 17539574, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3226, 8388601, F3, 23) (dual of [8388601, 8388375, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3226, large, F3, 23) (dual of [large, large−226, 24]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,22], and designed minimum distance d ≥ |I|+1 = 24 [i]
- discarding factors / shortening the dual code based on linear OA(3226, large, F3, 23) (dual of [large, large−226, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3226, 8388601, F3, 23) (dual of [8388601, 8388375, 24]-code), using
- net defined by OOA [i] based on linear OOA(3226, 762600, F3, 23, 23) (dual of [(762600, 23), 17539574, 24]-NRT-code), using
(212, 212+23, 2655120)-Net over F3 — Digital
Digital (212, 235, 2655120)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3235, 2655120, F3, 3, 23) (dual of [(2655120, 3), 7965125, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3235, 2796204, F3, 3, 23) (dual of [(2796204, 3), 8388377, 24]-NRT-code), using
- 3 times NRT-code embedding in larger space [i] based on linear OOA(3226, 2796201, F3, 3, 23) (dual of [(2796201, 3), 8388377, 24]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3226, large, F3, 23) (dual of [large, large−226, 24]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,22], and designed minimum distance d ≥ |I|+1 = 24 [i]
- OOA 3-folding [i] based on linear OA(3226, large, F3, 23) (dual of [large, large−226, 24]-code), using
- 3 times NRT-code embedding in larger space [i] based on linear OOA(3226, 2796201, F3, 3, 23) (dual of [(2796201, 3), 8388377, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3235, 2796204, F3, 3, 23) (dual of [(2796204, 3), 8388377, 24]-NRT-code), using
(212, 212+23, large)-Net in Base 3 — Upper bound on s
There is no (212, 235, large)-net in base 3, because
- 21 times m-reduction [i] would yield (212, 214, large)-net in base 3, but