Best Known (90, 90+13, s)-Nets in Base 3
(90, 90+13, 88578)-Net over F3 — Constructive and digital
Digital (90, 103, 88578)-net over F3, using
- 32 times duplication [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
(90, 90+13, 177157)-Net over F3 — Digital
Digital (90, 103, 177157)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3103, 177157, F3, 3, 13) (dual of [(177157, 3), 531368, 14]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3103, 531471, F3, 13) (dual of [531471, 531368, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(3103, 531472, F3, 13) (dual of [531472, 531369, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- linear OA(397, 531442, F3, 13) (dual of [531442, 531345, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 324−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(373, 531442, F3, 9) (dual of [531442, 531369, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 324−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(36, 30, F3, 3) (dual of [30, 24, 4]-code or 30-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3103, 531472, F3, 13) (dual of [531472, 531369, 14]-code), using
- OOA 3-folding [i] based on linear OA(3103, 531471, F3, 13) (dual of [531471, 531368, 14]-code), using
(90, 90+13, large)-Net in Base 3 — Upper bound on s
There is no (90, 103, large)-net in base 3, because
- 11 times m-reduction [i] would yield (90, 92, large)-net in base 3, but