Best Known (78−14, 78, s)-Nets in Base 9
(78−14, 78, 75924)-Net over F9 — Constructive and digital
Digital (64, 78, 75924)-net over F9, using
- 91 times duplication [i] based on digital (63, 77, 75924)-net over F9, using
- net defined by OOA [i] based on linear OOA(977, 75924, F9, 14, 14) (dual of [(75924, 14), 1062859, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(977, 531468, F9, 14) (dual of [531468, 531391, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(977, 531469, F9, 14) (dual of [531469, 531392, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- linear OA(973, 531441, F9, 14) (dual of [531441, 531368, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(949, 531441, F9, 10) (dual of [531441, 531392, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(94, 28, F9, 3) (dual of [28, 24, 4]-code or 28-cap in PG(3,9)), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(977, 531469, F9, 14) (dual of [531469, 531392, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(977, 531468, F9, 14) (dual of [531468, 531391, 15]-code), using
- net defined by OOA [i] based on linear OOA(977, 75924, F9, 14, 14) (dual of [(75924, 14), 1062859, 15]-NRT-code), using
(78−14, 78, 531471)-Net over F9 — Digital
Digital (64, 78, 531471)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(978, 531471, F9, 14) (dual of [531471, 531393, 15]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(977, 531469, F9, 14) (dual of [531469, 531392, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- linear OA(973, 531441, F9, 14) (dual of [531441, 531368, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(949, 531441, F9, 10) (dual of [531441, 531392, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(94, 28, F9, 3) (dual of [28, 24, 4]-code or 28-cap in PG(3,9)), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- linear OA(977, 531470, F9, 13) (dual of [531470, 531393, 14]-code), using Gilbert–Varšamov bound and bm = 977 > Vbs−1(k−1) = 72849 307019 693718 783445 670310 328948 160147 005687 266098 586602 730314 719209 [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(977, 531469, F9, 14) (dual of [531469, 531392, 15]-code), using
- construction X with Varšamov bound [i] based on
(78−14, 78, large)-Net in Base 9 — Upper bound on s
There is no (64, 78, large)-net in base 9, because
- 12 times m-reduction [i] would yield (64, 66, large)-net in base 9, but