Best Known (69−9, 69, s)-Nets in Base 3
(69−9, 69, 44293)-Net over F3 — Constructive and digital
Digital (60, 69, 44293)-net over F3, using
- net defined by OOA [i] based on linear OOA(369, 44293, F3, 9, 9) (dual of [(44293, 9), 398568, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(369, 177173, F3, 9) (dual of [177173, 177104, 10]-code), using
- 1 times code embedding in larger space [i] based on linear OA(368, 177172, F3, 9) (dual of [177172, 177104, 10]-code), using
- construction X4 applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(367, 177148, F3, 9) (dual of [177148, 177081, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(345, 177148, F3, 7) (dual of [177148, 177103, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(323, 24, F3, 23) (dual of [24, 1, 24]-code or 24-arc in PG(22,3)), using
- dual of repetition code with length 24 [i]
- linear OA(31, 24, F3, 1) (dual of [24, 23, 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
- 1 times code embedding in larger space [i] based on linear OA(368, 177172, F3, 9) (dual of [177172, 177104, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(369, 177173, F3, 9) (dual of [177173, 177104, 10]-code), using
(69−9, 69, 88586)-Net over F3 — Digital
Digital (60, 69, 88586)-net over F3, using
- 31 times duplication [i] based on digital (59, 68, 88586)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(368, 88586, F3, 2, 9) (dual of [(88586, 2), 177104, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(368, 177172, F3, 9) (dual of [177172, 177104, 10]-code), using
- construction X4 applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(367, 177148, F3, 9) (dual of [177148, 177081, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(345, 177148, F3, 7) (dual of [177148, 177103, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(323, 24, F3, 23) (dual of [24, 1, 24]-code or 24-arc in PG(22,3)), using
- dual of repetition code with length 24 [i]
- linear OA(31, 24, F3, 1) (dual of [24, 23, 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(368, 177172, F3, 9) (dual of [177172, 177104, 10]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(368, 88586, F3, 2, 9) (dual of [(88586, 2), 177104, 10]-NRT-code), using
(69−9, 69, large)-Net in Base 3 — Upper bound on s
There is no (60, 69, large)-net in base 3, because
- 7 times m-reduction [i] would yield (60, 62, large)-net in base 3, but