Best Known (59, 59+20, s)-Nets in Base 9
(59, 59+20, 1312)-Net over F9 — Constructive and digital
Digital (59, 79, 1312)-net over F9, using
- 91 times duplication [i] based on digital (58, 78, 1312)-net over F9, using
- net defined by OOA [i] based on linear OOA(978, 1312, F9, 20, 20) (dual of [(1312, 20), 26162, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(978, 13120, F9, 20) (dual of [13120, 13042, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(978, 13122, F9, 20) (dual of [13122, 13044, 21]-code), using
- trace code [i] based on linear OA(8139, 6561, F81, 20) (dual of [6561, 6522, 21]-code), using
- an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- trace code [i] based on linear OA(8139, 6561, F81, 20) (dual of [6561, 6522, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(978, 13122, F9, 20) (dual of [13122, 13044, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(978, 13120, F9, 20) (dual of [13120, 13042, 21]-code), using
- net defined by OOA [i] based on linear OOA(978, 1312, F9, 20, 20) (dual of [(1312, 20), 26162, 21]-NRT-code), using
(59, 59+20, 12875)-Net over F9 — Digital
Digital (59, 79, 12875)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(979, 12875, F9, 20) (dual of [12875, 12796, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(979, 13127, F9, 20) (dual of [13127, 13048, 21]-code), using
- 1 times code embedding in larger space [i] based on linear OA(978, 13126, F9, 20) (dual of [13126, 13048, 21]-code), using
- trace code [i] based on linear OA(8139, 6563, F81, 20) (dual of [6563, 6524, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(8139, 6561, F81, 20) (dual of [6561, 6522, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(8137, 6561, F81, 19) (dual of [6561, 6524, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(810, 2, F81, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(810, s, F81, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- trace code [i] based on linear OA(8139, 6563, F81, 20) (dual of [6563, 6524, 21]-code), using
- 1 times code embedding in larger space [i] based on linear OA(978, 13126, F9, 20) (dual of [13126, 13048, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(979, 13127, F9, 20) (dual of [13127, 13048, 21]-code), using
(59, 59+20, large)-Net in Base 9 — Upper bound on s
There is no (59, 79, large)-net in base 9, because
- 18 times m-reduction [i] would yield (59, 61, large)-net in base 9, but