Best Known (87−10, 87, s)-Nets in Base 3
(87−10, 87, 956597)-Net over F3 — Constructive and digital
Digital (77, 87, 956597)-net over F3, using
- 31 times duplication [i] based on digital (76, 86, 956597)-net over F3, using
- net defined by OOA [i] based on linear OOA(386, 956597, F3, 10, 10) (dual of [(956597, 10), 9565884, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(386, 4782985, F3, 10) (dual of [4782985, 4782899, 11]-code), using
- construction X4 applied to Ce(9) ⊂ Ce(7) [i] based on
- 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(371, 4782969, F3, 8) (dual of [4782969, 4782898, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(9) ⊂ Ce(7) [i] based on
- OA 5-folding and stacking [i] based on linear OA(386, 4782985, F3, 10) (dual of [4782985, 4782899, 11]-code), using
- net defined by OOA [i] based on linear OOA(386, 956597, F3, 10, 10) (dual of [(956597, 10), 9565884, 11]-NRT-code), using
(87−10, 87, 1594328)-Net over F3 — Digital
Digital (77, 87, 1594328)-net over F3, using
- 31 times duplication [i] based on digital (76, 86, 1594328)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(386, 1594328, F3, 3, 10) (dual of [(1594328, 3), 4782898, 11]-NRT-code), using
- OOA 3-folding [i] based on linear OA(386, 4782984, F3, 10) (dual of [4782984, 4782898, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- 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(371, 4782969, F3, 8) (dual of [4782969, 4782898, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(31, 15, F3, 1) (dual of [15, 14, 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 X applied to Ce(9) ⊂ Ce(7) [i] based on
- OOA 3-folding [i] based on linear OA(386, 4782984, F3, 10) (dual of [4782984, 4782898, 11]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(386, 1594328, F3, 3, 10) (dual of [(1594328, 3), 4782898, 11]-NRT-code), using
(87−10, 87, large)-Net in Base 3 — Upper bound on s
There is no (77, 87, large)-net in base 3, because
- 8 times m-reduction [i] would yield (77, 79, large)-net in base 3, but