Best Known (76, 76+17, s)-Nets in Base 3
(76, 76+17, 822)-Net over F3 — Constructive and digital
Digital (76, 93, 822)-net over F3, using
- net defined by OOA [i] based on linear OOA(393, 822, F3, 17, 17) (dual of [(822, 17), 13881, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(393, 6577, F3, 17) (dual of [6577, 6484, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(393, 6581, F3, 17) (dual of [6581, 6488, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- linear OA(389, 6561, F3, 17) (dual of [6561, 6472, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(373, 6561, F3, 14) (dual of [6561, 6488, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(16) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(393, 6581, F3, 17) (dual of [6581, 6488, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(393, 6577, F3, 17) (dual of [6577, 6484, 18]-code), using
(76, 76+17, 3290)-Net over F3 — Digital
Digital (76, 93, 3290)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(393, 3290, F3, 2, 17) (dual of [(3290, 2), 6487, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(393, 6580, F3, 17) (dual of [6580, 6487, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(393, 6581, F3, 17) (dual of [6581, 6488, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- linear OA(389, 6561, F3, 17) (dual of [6561, 6472, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(373, 6561, F3, 14) (dual of [6561, 6488, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(16) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(393, 6581, F3, 17) (dual of [6581, 6488, 18]-code), using
- OOA 2-folding [i] based on linear OA(393, 6580, F3, 17) (dual of [6580, 6487, 18]-code), using
(76, 76+17, 577495)-Net in Base 3 — Upper bound on s
There is no (76, 93, 577496)-net in base 3, because
- 1 times m-reduction [i] would yield (76, 92, 577496)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 78 551848 023782 440035 407087 964280 988759 111937 > 392 [i]