Best Known (90, 90+27, s)-Nets in Base 9
(90, 90+27, 1012)-Net over F9 — Constructive and digital
Digital (90, 117, 1012)-net over F9, using
- 91 times duplication [i] based on digital (89, 116, 1012)-net over F9, using
- net defined by OOA [i] based on linear OOA(9116, 1012, F9, 27, 27) (dual of [(1012, 27), 27208, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(9116, 13157, F9, 27) (dual of [13157, 13041, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(9116, 13158, F9, 27) (dual of [13158, 13042, 28]-code), using
- trace code [i] based on linear OA(8158, 6579, F81, 27) (dual of [6579, 6521, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,10]) [i] based on
- linear OA(8153, 6562, F81, 27) (dual of [6562, 6509, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(8141, 6562, F81, 21) (dual of [6562, 6521, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(815, 17, F81, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,81)), using
- discarding factors / shortening the dual code based on linear OA(815, 81, F81, 5) (dual of [81, 76, 6]-code or 81-arc in PG(4,81)), using
- Reed–Solomon code RS(76,81) [i]
- discarding factors / shortening the dual code based on linear OA(815, 81, F81, 5) (dual of [81, 76, 6]-code or 81-arc in PG(4,81)), using
- construction X applied to C([0,13]) ⊂ C([0,10]) [i] based on
- trace code [i] based on linear OA(8158, 6579, F81, 27) (dual of [6579, 6521, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(9116, 13158, F9, 27) (dual of [13158, 13042, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(9116, 13157, F9, 27) (dual of [13157, 13041, 28]-code), using
- net defined by OOA [i] based on linear OOA(9116, 1012, F9, 27, 27) (dual of [(1012, 27), 27208, 28]-NRT-code), using
(90, 90+27, 1513)-Net in Base 9 — Constructive
(90, 117, 1513)-net in base 9, using
- base change [i] based on digital (51, 78, 1513)-net over F27, using
- net defined by OOA [i] based on linear OOA(2778, 1513, F27, 27, 27) (dual of [(1513, 27), 40773, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2778, 19670, F27, 27) (dual of [19670, 19592, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2778, 19682, F27, 27) (dual of [19682, 19604, 28]-code), using
- 1 times truncation [i] based on linear OA(2779, 19683, F27, 28) (dual of [19683, 19604, 29]-code), using
- an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- 1 times truncation [i] based on linear OA(2779, 19683, F27, 28) (dual of [19683, 19604, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(2778, 19682, F27, 27) (dual of [19682, 19604, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2778, 19670, F27, 27) (dual of [19670, 19592, 28]-code), using
- net defined by OOA [i] based on linear OOA(2778, 1513, F27, 27, 27) (dual of [(1513, 27), 40773, 28]-NRT-code), using
(90, 90+27, 25972)-Net over F9 — Digital
Digital (90, 117, 25972)-net over F9, using
(90, 90+27, large)-Net in Base 9 — Upper bound on s
There is no (90, 117, large)-net in base 9, because
- 25 times m-reduction [i] would yield (90, 92, large)-net in base 9, but