Best Known (51−12, 51, s)-Nets in Base 9
(51−12, 51, 9842)-Net over F9 — Constructive and digital
Digital (39, 51, 9842)-net over F9, using
- net defined by OOA [i] based on linear OOA(951, 9842, F9, 12, 12) (dual of [(9842, 12), 118053, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(951, 59052, F9, 12) (dual of [59052, 59001, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(951, 59054, F9, 12) (dual of [59054, 59003, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(951, 59049, F9, 12) (dual of [59049, 58998, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(946, 59049, F9, 11) (dual of [59049, 59003, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(90, 5, F9, 0) (dual of [5, 5, 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
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(951, 59054, F9, 12) (dual of [59054, 59003, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(951, 59052, F9, 12) (dual of [59052, 59001, 13]-code), using
(51−12, 51, 33422)-Net over F9 — Digital
Digital (39, 51, 33422)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(951, 33422, F9, 12) (dual of [33422, 33371, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(951, 59049, F9, 12) (dual of [59049, 58998, 13]-code), using
- an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- discarding factors / shortening the dual code based on linear OA(951, 59049, F9, 12) (dual of [59049, 58998, 13]-code), using
(51−12, 51, large)-Net in Base 9 — Upper bound on s
There is no (39, 51, large)-net in base 9, because
- 10 times m-reduction [i] would yield (39, 41, large)-net in base 9, but