Best Known (68−12, 68, s)-Nets in Base 3
(68−12, 68, 1096)-Net over F3 — Constructive and digital
Digital (56, 68, 1096)-net over F3, using
- 1 times m-reduction [i] based on digital (56, 69, 1096)-net over F3, using
- net defined by OOA [i] based on linear OOA(369, 1096, F3, 13, 13) (dual of [(1096, 13), 14179, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(369, 6577, F3, 13) (dual of [6577, 6508, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(369, 6581, F3, 13) (dual of [6581, 6512, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(365, 6561, F3, 13) (dual of [6561, 6496, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(349, 6561, F3, 10) (dual of [6561, 6512, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(34, 20, F3, 2) (dual of [20, 16, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(369, 6581, F3, 13) (dual of [6581, 6512, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(369, 6577, F3, 13) (dual of [6577, 6508, 14]-code), using
- net defined by OOA [i] based on linear OOA(369, 1096, F3, 13, 13) (dual of [(1096, 13), 14179, 14]-NRT-code), using
(68−12, 68, 3553)-Net over F3 — Digital
Digital (56, 68, 3553)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(368, 3553, F3, 12) (dual of [3553, 3485, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(368, 6575, F3, 12) (dual of [6575, 6507, 13]-code), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- linear OA(365, 6562, F3, 13) (dual of [6562, 6497, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(349, 6562, F3, 9) (dual of [6562, 6513, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(368, 6575, F3, 12) (dual of [6575, 6507, 13]-code), using
(68−12, 68, 382437)-Net in Base 3 — Upper bound on s
There is no (56, 68, 382438)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 278 132142 788281 040918 284067 632149 > 368 [i]