Best Known (109, 109+15, s)-Nets in Base 3
(109, 109+15, 75924)-Net over F3 — Constructive and digital
Digital (109, 124, 75924)-net over F3, using
- 31 times duplication [i] based on digital (108, 123, 75924)-net over F3, using
- net defined by OOA [i] based on linear OOA(3123, 75924, F3, 15, 15) (dual of [(75924, 15), 1138737, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3123, 531469, F3, 15) (dual of [531469, 531346, 16]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3122, 531468, F3, 15) (dual of [531468, 531346, 16]-code), using
- construction X4 applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(3121, 531442, F3, 15) (dual of [531442, 531321, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 324−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(397, 531442, F3, 13) (dual of [531442, 531345, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 324−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(325, 26, F3, 25) (dual of [26, 1, 26]-code or 26-arc in PG(24,3)), using
- dual of repetition code with length 26 [i]
- linear OA(31, 26, F3, 1) (dual of [26, 25, 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([0,7]) ⊂ C([0,6]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3122, 531468, F3, 15) (dual of [531468, 531346, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3123, 531469, F3, 15) (dual of [531469, 531346, 16]-code), using
- net defined by OOA [i] based on linear OOA(3123, 75924, F3, 15, 15) (dual of [(75924, 15), 1138737, 16]-NRT-code), using
(109, 109+15, 187516)-Net over F3 — Digital
Digital (109, 124, 187516)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3124, 187516, F3, 2, 15) (dual of [(187516, 2), 374908, 16]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3124, 265735, F3, 2, 15) (dual of [(265735, 2), 531346, 16]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3122, 265734, F3, 2, 15) (dual of [(265734, 2), 531346, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3122, 531468, F3, 15) (dual of [531468, 531346, 16]-code), using
- construction X4 applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(3121, 531442, F3, 15) (dual of [531442, 531321, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 324−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(397, 531442, F3, 13) (dual of [531442, 531345, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 324−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(325, 26, F3, 25) (dual of [26, 1, 26]-code or 26-arc in PG(24,3)), using
- dual of repetition code with length 26 [i]
- linear OA(31, 26, F3, 1) (dual of [26, 25, 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([0,7]) ⊂ C([0,6]) [i] based on
- OOA 2-folding [i] based on linear OA(3122, 531468, F3, 15) (dual of [531468, 531346, 16]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3122, 265734, F3, 2, 15) (dual of [(265734, 2), 531346, 16]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3124, 265735, F3, 2, 15) (dual of [(265735, 2), 531346, 16]-NRT-code), using
(109, 109+15, large)-Net in Base 3 — Upper bound on s
There is no (109, 124, large)-net in base 3, because
- 13 times m-reduction [i] would yield (109, 111, large)-net in base 3, but