Best Known (42−12, 42, s)-Nets in Base 9
(42−12, 42, 1095)-Net over F9 — Constructive and digital
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
(42−12, 42, 4622)-Net over F9 — Digital
Digital (30, 42, 4622)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(942, 4622, F9, 12) (dual of [4622, 4580, 13]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(942, 6570, F9, 12) (dual of [6570, 6528, 13]-code), using
(42−12, 42, 1789900)-Net in Base 9 — Upper bound on s
There is no (30, 42, 1789901)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 11972 537912 853939 365948 358836 761325 665201 > 942 [i]