Best Known (233−19, 233, s)-Nets in Base 3
(233−19, 233, 1062885)-Net over F3 — Constructive and digital
Digital (214, 233, 1062885)-net over F3, using
- 31 times duplication [i] based on digital (213, 232, 1062885)-net over F3, using
- net defined by OOA [i] based on linear OOA(3232, 1062885, F3, 21, 19) (dual of [(1062885, 21), 22320353, 20]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(3232, 3188656, F3, 3, 19) (dual of [(3188656, 3), 9565736, 20]-NRT-code), using
- trace code [i] based on linear OOA(9116, 1594328, F9, 3, 19) (dual of [(1594328, 3), 4782868, 20]-NRT-code), using
- OOA 3-folding [i] based on linear OA(9116, 4782984, F9, 19) (dual of [4782984, 4782868, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(9116, 4782986, F9, 19) (dual of [4782986, 4782870, 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(93, 17, F9, 2) (dual of [17, 14, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(93, 80, F9, 2) (dual of [80, 77, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 92−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(93, 80, F9, 2) (dual of [80, 77, 3]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(9116, 4782986, F9, 19) (dual of [4782986, 4782870, 20]-code), using
- OOA 3-folding [i] based on linear OA(9116, 4782984, F9, 19) (dual of [4782984, 4782868, 20]-code), using
- trace code [i] based on linear OOA(9116, 1594328, F9, 3, 19) (dual of [(1594328, 3), 4782868, 20]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(3232, 3188656, F3, 3, 19) (dual of [(3188656, 3), 9565736, 20]-NRT-code), using
- net defined by OOA [i] based on linear OOA(3232, 1062885, F3, 21, 19) (dual of [(1062885, 21), 22320353, 20]-NRT-code), using
(233−19, 233, large)-Net over F3 — Digital
Digital (214, 233, large)-net over F3, using
- 35 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
(233−19, 233, large)-Net in Base 3 — Upper bound on s
There is no (214, 233, large)-net in base 3, because
- 17 times m-reduction [i] would yield (214, 216, large)-net in base 3, but