Best Known (12, 12+4, s)-Nets in Base 9
(12, 12+4, 29527)-Net over F9 — Constructive and digital
Digital (12, 16, 29527)-net over F9, using
- net defined by OOA [i] based on linear OOA(916, 29527, F9, 4, 4) (dual of [(29527, 4), 118092, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(916, 29527, F9, 3, 4) (dual of [(29527, 3), 88565, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(916, 59054, F9, 4) (dual of [59054, 59038, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(916, 59049, F9, 4) (dual of [59049, 59033, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(911, 59049, F9, 3) (dual of [59049, 59038, 4]-code or 59049-cap in PG(10,9)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [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(3) ⊂ Ce(2) [i] based on
- OA 2-folding and stacking [i] based on linear OA(916, 59054, F9, 4) (dual of [59054, 59038, 5]-code), using
- appending kth column [i] based on linear OOA(916, 29527, F9, 3, 4) (dual of [(29527, 3), 88565, 5]-NRT-code), using
(12, 12+4, 59054)-Net over F9 — Digital
Digital (12, 16, 59054)-net over F9, using
- net defined by OOA [i] based on linear OOA(916, 59054, F9, 4, 4) (dual of [(59054, 4), 236200, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(916, 59054, F9, 3, 4) (dual of [(59054, 3), 177146, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(916, 59054, F9, 4) (dual of [59054, 59038, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(916, 59049, F9, 4) (dual of [59049, 59033, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(911, 59049, F9, 3) (dual of [59049, 59038, 4]-code or 59049-cap in PG(10,9)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [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(3) ⊂ Ce(2) [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(916, 59054, F9, 4) (dual of [59054, 59038, 5]-code), using
- appending kth column [i] based on linear OOA(916, 59054, F9, 3, 4) (dual of [(59054, 3), 177146, 5]-NRT-code), using
(12, 12+4, 7609656)-Net in Base 9 — Upper bound on s
There is no (12, 16, 7609657)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 1853 020514 308305 > 916 [i]