Best Known (86−9, 86, s)-Nets in Base 3
(86−9, 86, 1195749)-Net over F3 — Constructive and digital
Digital (77, 86, 1195749)-net over F3, using
- net defined by OOA [i] based on linear OOA(386, 1195749, F3, 9, 9) (dual of [(1195749, 9), 10761655, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(386, 4782997, F3, 9) (dual of [4782997, 4782911, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(386, 4782999, F3, 9) (dual of [4782999, 4782913, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(385, 4782970, F3, 9) (dual of [4782970, 4782885, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 328−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(357, 4782970, F3, 7) (dual of [4782970, 4782913, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 328−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(31, 29, F3, 1) (dual of [29, 28, 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 C([0,4]) ⊂ C([0,3]) [i] based on
- discarding factors / shortening the dual code based on linear OA(386, 4782999, F3, 9) (dual of [4782999, 4782913, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(386, 4782997, F3, 9) (dual of [4782997, 4782911, 10]-code), using
(86−9, 86, 2391500)-Net over F3 — Digital
Digital (77, 86, 2391500)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(386, 2391500, F3, 2, 9) (dual of [(2391500, 2), 4782914, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(386, 4783000, F3, 9) (dual of [4783000, 4782914, 10]-code), using
- construction X4 applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(385, 4782970, F3, 9) (dual of [4782970, 4782885, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 328−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(357, 4782970, F3, 7) (dual of [4782970, 4782913, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 328−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(329, 30, F3, 29) (dual of [30, 1, 30]-code or 30-arc in PG(28,3)), using
- dual of repetition code with length 30 [i]
- linear OA(31, 30, F3, 1) (dual of [30, 29, 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 C([0,4]) ⊂ C([0,3]) [i] based on
- OOA 2-folding [i] based on linear OA(386, 4783000, F3, 9) (dual of [4783000, 4782914, 10]-code), using
(86−9, 86, large)-Net in Base 3 — Upper bound on s
There is no (77, 86, large)-net in base 3, because
- 7 times m-reduction [i] would yield (77, 79, large)-net in base 3, but