Best Known (40, 52, s)-Nets in Base 9
(40, 52, 9843)-Net over F9 — Constructive and digital
Digital (40, 52, 9843)-net over F9, using
- net defined by OOA [i] based on linear OOA(952, 9843, F9, 12, 12) (dual of [(9843, 12), 118064, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(952, 59058, F9, 12) (dual of [59058, 59006, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(952, 59060, F9, 12) (dual of [59060, 59008, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [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(941, 59049, F9, 10) (dual of [59049, 59008, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(91, 11, F9, 1) (dual of [11, 10, 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(952, 59060, F9, 12) (dual of [59060, 59008, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(952, 59058, F9, 12) (dual of [59058, 59006, 13]-code), using
(40, 52, 41636)-Net over F9 — Digital
Digital (40, 52, 41636)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(952, 41636, F9, 12) (dual of [41636, 41584, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(952, 59060, F9, 12) (dual of [59060, 59008, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [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(941, 59049, F9, 10) (dual of [59049, 59008, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(91, 11, F9, 1) (dual of [11, 10, 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(952, 59060, F9, 12) (dual of [59060, 59008, 13]-code), using
(40, 52, large)-Net in Base 9 — Upper bound on s
There is no (40, 52, large)-net in base 9, because
- 10 times m-reduction [i] would yield (40, 42, large)-net in base 9, but