Best Known (77, 77+17, s)-Nets in Base 3
(77, 77+17, 822)-Net over F3 — Constructive and digital
Digital (77, 94, 822)-net over F3, using
- 31 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(393, 822, F3, 17, 17) (dual of [(822, 17), 13881, 18]-NRT-code), using
(77, 77+17, 3291)-Net over F3 — Digital
Digital (77, 94, 3291)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(394, 3291, F3, 2, 17) (dual of [(3291, 2), 6488, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(394, 6582, F3, 17) (dual of [6582, 6488, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(394, 6583, F3, 17) (dual of [6583, 6489, 18]-code), using
- construction XX applied to Ce(16) ⊂ Ce(13) ⊂ Ce(12) [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(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(34, 21, F3, 2) (dual of [21, 17, 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
- linear OA(30, 1, F3, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(16) ⊂ Ce(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(394, 6583, F3, 17) (dual of [6583, 6489, 18]-code), using
- OOA 2-folding [i] based on linear OA(394, 6582, F3, 17) (dual of [6582, 6488, 18]-code), using
(77, 77+17, 662505)-Net in Base 3 — Upper bound on s
There is no (77, 94, 662506)-net in base 3, because
- 1 times m-reduction [i] would yield (77, 93, 662506)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 235 655347 932693 886969 412164 271871 317129 468625 > 393 [i]