Best Known (22−4, 22, s)-Nets in Base 9
(22−4, 22, 2391488)-Net over F9 — Constructive and digital
Digital (18, 22, 2391488)-net over F9, using
- net defined by OOA [i] based on linear OOA(922, 2391488, F9, 4, 4) (dual of [(2391488, 4), 9565930, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(922, 2391488, F9, 3, 4) (dual of [(2391488, 3), 7174442, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(922, 4782976, F9, 4) (dual of [4782976, 4782954, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(922, 4782969, F9, 4) (dual of [4782969, 4782947, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(915, 4782969, F9, 3) (dual of [4782969, 4782954, 4]-code or 4782969-cap in PG(14,9)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(90, 7, F9, 0) (dual of [7, 7, 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(922, 4782976, F9, 4) (dual of [4782976, 4782954, 5]-code), using
- appending kth column [i] based on linear OOA(922, 2391488, F9, 3, 4) (dual of [(2391488, 3), 7174442, 5]-NRT-code), using
(22−4, 22, 4782976)-Net over F9 — Digital
Digital (18, 22, 4782976)-net over F9, using
- net defined by OOA [i] based on linear OOA(922, 4782976, F9, 4, 4) (dual of [(4782976, 4), 19131882, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(922, 4782976, F9, 3, 4) (dual of [(4782976, 3), 14348906, 5]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(922, 4782976, F9, 4) (dual of [4782976, 4782954, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(922, 4782969, F9, 4) (dual of [4782969, 4782947, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(915, 4782969, F9, 3) (dual of [4782969, 4782954, 4]-code or 4782969-cap in PG(14,9)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(90, 7, F9, 0) (dual of [7, 7, 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(922, 4782976, F9, 4) (dual of [4782976, 4782954, 5]-code), using
- appending kth column [i] based on linear OOA(922, 4782976, F9, 3, 4) (dual of [(4782976, 3), 14348906, 5]-NRT-code), using
(22−4, 22, large)-Net in Base 9 — Upper bound on s
There is no (18, 22, large)-net in base 9, because
- 2 times m-reduction [i] would yield (18, 20, large)-net in base 9, but