Best Known (52, 52+14, s)-Nets in Base 9
(52, 52+14, 8439)-Net over F9 — Constructive and digital
Digital (52, 66, 8439)-net over F9, using
- 91 times duplication [i] based on digital (51, 65, 8439)-net over F9, using
- net defined by OOA [i] based on linear OOA(965, 8439, F9, 14, 14) (dual of [(8439, 14), 118081, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(965, 59073, F9, 14) (dual of [59073, 59008, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- linear OA(961, 59049, F9, 14) (dual of [59049, 58988, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(941, 59049, F9, 10) (dual of [59049, 59008, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(94, 24, F9, 3) (dual of [24, 20, 4]-code or 24-cap in PG(3,9)), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- OA 7-folding and stacking [i] based on linear OA(965, 59073, F9, 14) (dual of [59073, 59008, 15]-code), using
- net defined by OOA [i] based on linear OOA(965, 8439, F9, 14, 14) (dual of [(8439, 14), 118081, 15]-NRT-code), using
(52, 52+14, 59075)-Net over F9 — Digital
Digital (52, 66, 59075)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(966, 59075, F9, 14) (dual of [59075, 59009, 15]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(965, 59073, F9, 14) (dual of [59073, 59008, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- linear OA(961, 59049, F9, 14) (dual of [59049, 58988, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(941, 59049, F9, 10) (dual of [59049, 59008, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(94, 24, F9, 3) (dual of [24, 20, 4]-code or 24-cap in PG(3,9)), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- linear OA(965, 59074, F9, 13) (dual of [59074, 59009, 14]-code), using Gilbert–Varšamov bound and bm = 965 > Vbs−1(k−1) = 258783 737228 190251 608001 279041 810027 697777 298202 466447 576585 [i]
- linear OA(90, 1, F9, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(965, 59073, F9, 14) (dual of [59073, 59008, 15]-code), using
- construction X with Varšamov bound [i] based on
(52, 52+14, large)-Net in Base 9 — Upper bound on s
There is no (52, 66, large)-net in base 9, because
- 12 times m-reduction [i] would yield (52, 54, large)-net in base 9, but