Best Known (103−17, 103, s)-Nets in Base 3
(103−17, 103, 2462)-Net over F3 — Constructive and digital
Digital (86, 103, 2462)-net over F3, using
- net defined by OOA [i] based on linear OOA(3103, 2462, F3, 17, 17) (dual of [(2462, 17), 41751, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3103, 19697, F3, 17) (dual of [19697, 19594, 18]-code), using
- construction XX applied to Ce(16) ⊂ Ce(15) ⊂ Ce(13) [i] based on
- linear OA(3100, 19683, F3, 17) (dual of [19683, 19583, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(391, 19683, F3, 16) (dual of [19683, 19592, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(382, 19683, F3, 14) (dual of [19683, 19601, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(30, 11, F3, 0) (dual of [11, 11, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(31, 3, F3, 1) (dual of [3, 2, 2]-code), using
- Reed–Solomon code RS(2,3) [i]
- construction XX applied to Ce(16) ⊂ Ce(15) ⊂ Ce(13) [i] based on
- OOA 8-folding and stacking with additional row [i] based on linear OA(3103, 19697, F3, 17) (dual of [19697, 19594, 18]-code), using
(103−17, 103, 8353)-Net over F3 — Digital
Digital (86, 103, 8353)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3103, 8353, F3, 2, 17) (dual of [(8353, 2), 16603, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3103, 9848, F3, 2, 17) (dual of [(9848, 2), 19593, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3103, 19696, F3, 17) (dual of [19696, 19593, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- linear OA(3100, 19683, F3, 17) (dual of [19683, 19583, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(382, 19683, F3, 14) (dual of [19683, 19601, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- OOA 2-folding [i] based on linear OA(3103, 19696, F3, 17) (dual of [19696, 19593, 18]-code), using
- discarding factors / shortening the dual code based on linear OOA(3103, 9848, F3, 2, 17) (dual of [(9848, 2), 19593, 18]-NRT-code), using
(103−17, 103, 2280103)-Net in Base 3 — Upper bound on s
There is no (86, 103, 2280104)-net in base 3, because
- 1 times m-reduction [i] would yield (86, 102, 2280104)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 4 638399 187110 147855 249279 232660 246154 446244 850945 > 3102 [i]