Best Known (53−14, 53, s)-Nets in Base 9
(53−14, 53, 940)-Net over F9 — Constructive and digital
Digital (39, 53, 940)-net over F9, using
- net defined by OOA [i] based on linear OOA(953, 940, F9, 14, 14) (dual of [(940, 14), 13107, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(953, 6580, F9, 14) (dual of [6580, 6527, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(953, 6581, F9, 14) (dual of [6581, 6528, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- linear OA(949, 6561, F9, 14) (dual of [6561, 6512, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(933, 6561, F9, 10) (dual of [6561, 6528, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(94, 20, F9, 3) (dual of [20, 16, 4]-code or 20-cap in PG(3,9)), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(953, 6581, F9, 14) (dual of [6581, 6528, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(953, 6580, F9, 14) (dual of [6580, 6527, 15]-code), using
(53−14, 53, 6581)-Net over F9 — Digital
Digital (39, 53, 6581)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(953, 6581, F9, 14) (dual of [6581, 6528, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- linear OA(949, 6561, F9, 14) (dual of [6561, 6512, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(933, 6561, F9, 10) (dual of [6561, 6528, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(94, 20, F9, 3) (dual of [20, 16, 4]-code or 20-cap in PG(3,9)), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
(53−14, 53, 7092631)-Net in Base 9 — Upper bound on s
There is no (39, 53, 7092632)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 375 710232 590233 155652 441489 182172 564326 519536 435265 > 953 [i]