Best Known (31, 31+12, s)-Nets in Base 9
(31, 31+12, 1095)-Net over F9 — Constructive and digital
Digital (31, 43, 1095)-net over F9, using
- 91 times duplication [i] based on digital (30, 42, 1095)-net over F9, using
- net defined by OOA [i] based on linear OOA(942, 1095, F9, 12, 12) (dual of [(1095, 12), 13098, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(942, 6570, F9, 12) (dual of [6570, 6528, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- linear OA(941, 6561, F9, 12) (dual of [6561, 6520, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(91, 9, F9, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, s, F9, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- OA 6-folding and stacking [i] based on linear OA(942, 6570, F9, 12) (dual of [6570, 6528, 13]-code), using
- net defined by OOA [i] based on linear OOA(942, 1095, F9, 12, 12) (dual of [(1095, 12), 13098, 13]-NRT-code), using
(31, 31+12, 5759)-Net over F9 — Digital
Digital (31, 43, 5759)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(943, 5759, F9, 12) (dual of [5759, 5716, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(943, 6571, F9, 12) (dual of [6571, 6528, 13]-code), using
- 1 times code embedding in larger space [i] based on linear OA(942, 6570, F9, 12) (dual of [6570, 6528, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- linear OA(941, 6561, F9, 12) (dual of [6561, 6520, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(91, 9, F9, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, s, F9, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(942, 6570, F9, 12) (dual of [6570, 6528, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(943, 6571, F9, 12) (dual of [6571, 6528, 13]-code), using
(31, 31+12, 2581484)-Net in Base 9 — Upper bound on s
There is no (31, 43, 2581485)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 107752 744681 670770 781490 337271 155394 745777 > 943 [i]