Best Known (102−12, 102, s)-Nets in Base 3
(102−12, 102, 88578)-Net over F3 — Constructive and digital
Digital (90, 102, 88578)-net over F3, using
- 31 times duplication [i] based on digital (89, 101, 88578)-net over F3, using
- t-expansion [i] based on digital (88, 101, 88578)-net over F3, using
- net defined by OOA [i] based on linear OOA(3101, 88578, F3, 13, 13) (dual of [(88578, 13), 1151413, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3101, 531469, F3, 13) (dual of [531469, 531368, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(397, 531441, F3, 13) (dual of [531441, 531344, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(373, 531441, F3, 10) (dual of [531441, 531368, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(34, 28, F3, 2) (dual of [28, 24, 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
- OOA 6-folding and stacking with additional row [i] based on linear OA(3101, 531469, F3, 13) (dual of [531469, 531368, 14]-code), using
- net defined by OOA [i] based on linear OOA(3101, 88578, F3, 13, 13) (dual of [(88578, 13), 1151413, 14]-NRT-code), using
- t-expansion [i] based on digital (88, 101, 88578)-net over F3, using
(102−12, 102, 265736)-Net over F3 — Digital
Digital (90, 102, 265736)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3102, 265736, F3, 2, 12) (dual of [(265736, 2), 531370, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3102, 531472, F3, 12) (dual of [531472, 531370, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(3102, 531473, F3, 12) (dual of [531473, 531371, 13]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(398, 531467, F3, 12) (dual of [531467, 531369, 13]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(397, 531441, F3, 13) (dual of [531441, 531344, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(373, 531441, F3, 10) (dual of [531441, 531368, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(325, 26, F3, 25) (dual of [26, 1, 26]-code or 26-arc in PG(24,3)), using
- dual of repetition code with length 26 [i]
- linear OA(31, 26, F3, 1) (dual of [26, 25, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(398, 531469, F3, 9) (dual of [531469, 531371, 10]-code), using Gilbert–Varšamov bound and bm = 398 > Vbs−1(k−1) = 40 412600 174106 948585 131762 490626 303838 690289 [i]
- linear OA(32, 4, F3, 2) (dual of [4, 2, 3]-code or 4-arc in PG(1,3)), using
- extended Reed–Solomon code RSe(2,3) [i]
- Hamming code H(2,3) [i]
- Simplex code S(2,3) [i]
- the Tetracode [i]
- linear OA(398, 531467, F3, 12) (dual of [531467, 531369, 13]-code), using
- construction X with Varšamov bound [i] based on
- discarding factors / shortening the dual code based on linear OA(3102, 531473, F3, 12) (dual of [531473, 531371, 13]-code), using
- OOA 2-folding [i] based on linear OA(3102, 531472, F3, 12) (dual of [531472, 531370, 13]-code), using
(102−12, 102, large)-Net in Base 3 — Upper bound on s
There is no (90, 102, large)-net in base 3, because
- 10 times m-reduction [i] would yield (90, 92, large)-net in base 3, but