Best Known (89, 89+11, s)-Nets in Base 3
(89, 89+11, 956596)-Net over F3 — Constructive and digital
Digital (89, 100, 956596)-net over F3, using
- 31 times duplication [i] based on digital (88, 99, 956596)-net over F3, using
- net defined by OOA [i] based on linear OOA(399, 956596, F3, 11, 11) (dual of [(956596, 11), 10522457, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(399, 4782981, F3, 11) (dual of [4782981, 4782882, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(399, 4782983, F3, 11) (dual of [4782983, 4782884, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(399, 4782969, F3, 11) (dual of [4782969, 4782870, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(385, 4782969, F3, 10) (dual of [4782969, 4782884, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(30, 14, F3, 0) (dual of [14, 14, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(399, 4782983, F3, 11) (dual of [4782983, 4782884, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(399, 4782981, F3, 11) (dual of [4782981, 4782882, 12]-code), using
- net defined by OOA [i] based on linear OOA(399, 956596, F3, 11, 11) (dual of [(956596, 11), 10522457, 12]-NRT-code), using
(89, 89+11, 1594328)-Net over F3 — Digital
Digital (89, 100, 1594328)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3100, 1594328, F3, 3, 11) (dual of [(1594328, 3), 4782884, 12]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3100, 4782984, F3, 11) (dual of [4782984, 4782884, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(3100, 4782985, F3, 11) (dual of [4782985, 4782885, 12]-code), using
- construction X4 applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(399, 4782969, F3, 11) (dual of [4782969, 4782870, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(385, 4782969, F3, 10) (dual of [4782969, 4782884, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(315, 16, F3, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,3)), using
- dual of repetition code with length 16 [i]
- linear OA(31, 16, F3, 1) (dual of [16, 15, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(10) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(3100, 4782985, F3, 11) (dual of [4782985, 4782885, 12]-code), using
- OOA 3-folding [i] based on linear OA(3100, 4782984, F3, 11) (dual of [4782984, 4782884, 12]-code), using
(89, 89+11, large)-Net in Base 3 — Upper bound on s
There is no (89, 100, large)-net in base 3, because
- 9 times m-reduction [i] would yield (89, 91, large)-net in base 3, but