Best Known (118−18, 118, s)-Nets in Base 9
(118−18, 118, 531444)-Net over F9 — Constructive and digital
Digital (100, 118, 531444)-net over F9, using
- 1 times m-reduction [i] based on digital (100, 119, 531444)-net over F9, using
- net defined by OOA [i] based on linear OOA(9119, 531444, F9, 19, 19) (dual of [(531444, 19), 10097317, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(9119, 4782997, F9, 19) (dual of [4782997, 4782878, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(9119, 4783003, F9, 19) (dual of [4783003, 4782884, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(9113, 4782969, F9, 19) (dual of [4782969, 4782856, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(985, 4782969, F9, 14) (dual of [4782969, 4782884, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(96, 34, F9, 4) (dual of [34, 28, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(96, 72, F9, 4) (dual of [72, 66, 5]-code), using
- 1 times truncation [i] based on linear OA(97, 73, F9, 5) (dual of [73, 66, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(96, 72, F9, 4) (dual of [72, 66, 5]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(9119, 4783003, F9, 19) (dual of [4783003, 4782884, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(9119, 4782997, F9, 19) (dual of [4782997, 4782878, 20]-code), using
- net defined by OOA [i] based on linear OOA(9119, 531444, F9, 19, 19) (dual of [(531444, 19), 10097317, 20]-NRT-code), using
(118−18, 118, 4783003)-Net over F9 — Digital
Digital (100, 118, 4783003)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9118, 4783003, F9, 18) (dual of [4783003, 4782885, 19]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(9117, 4783001, F9, 18) (dual of [4783001, 4782884, 19]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(9113, 4782969, F9, 19) (dual of [4782969, 4782856, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(985, 4782969, F9, 14) (dual of [4782969, 4782884, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(94, 32, F9, 3) (dual of [32, 28, 4]-code or 32-cap in PG(3,9)), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(9117, 4783002, F9, 17) (dual of [4783002, 4782885, 18]-code), using Gilbert–Varšamov bound and bm = 9117 > Vbs−1(k−1) = 1 009292 463814 366584 286158 708107 182248 611649 524185 002343 876765 173304 388577 561308 203842 006942 323565 812557 656009 [i]
- linear OA(90, 1, F9, 0) (dual of [1, 1, 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
- linear OA(9117, 4783001, F9, 18) (dual of [4783001, 4782884, 19]-code), using
- construction X with Varšamov bound [i] based on
(118−18, 118, large)-Net in Base 9 — Upper bound on s
There is no (100, 118, large)-net in base 9, because
- 16 times m-reduction [i] would yield (100, 102, large)-net in base 9, but