Best Known (212, 212+19, s)-Nets in Base 3
(212, 212+19, 1062884)-Net over F3 — Constructive and digital
Digital (212, 231, 1062884)-net over F3, using
- 31 times duplication [i] based on digital (211, 230, 1062884)-net over F3, using
- trace code for nets [i] based on digital (96, 115, 531442)-net over F9, using
- net defined by OOA [i] based on linear OOA(9115, 531442, F9, 19, 19) (dual of [(531442, 19), 10097283, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(9115, 4782979, F9, 19) (dual of [4782979, 4782864, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(9113, 4782969, F9, 19) (dual of [4782969, 4782856, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(999, 4782969, F9, 16) (dual of [4782969, 4782870, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(92, 10, F9, 2) (dual of [10, 8, 3]-code or 10-arc in PG(1,9)), using
- extended Reed–Solomon code RSe(8,9) [i]
- Hamming code H(2,9) [i]
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- OOA 9-folding and stacking with additional row [i] based on linear OA(9115, 4782979, F9, 19) (dual of [4782979, 4782864, 20]-code), using
- net defined by OOA [i] based on linear OOA(9115, 531442, F9, 19, 19) (dual of [(531442, 19), 10097283, 20]-NRT-code), using
- trace code for nets [i] based on digital (96, 115, 531442)-net over F9, using
(212, 212+19, large)-Net over F3 — Digital
Digital (212, 231, large)-net over F3, using
- 33 times duplication [i] based on digital (209, 228, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3228, large, F3, 19) (dual of [large, large−228, 20]-code), using
- 47 times code embedding in larger space [i] based on linear OA(3181, large, F3, 19) (dual of [large, large−181, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 330−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- 47 times code embedding in larger space [i] based on linear OA(3181, large, F3, 19) (dual of [large, large−181, 20]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3228, large, F3, 19) (dual of [large, large−228, 20]-code), using
(212, 212+19, large)-Net in Base 3 — Upper bound on s
There is no (212, 231, large)-net in base 3, because
- 17 times m-reduction [i] would yield (212, 214, large)-net in base 3, but