Best Known (138, 138+21, s)-Nets in Base 3
(138, 138+21, 17717)-Net over F3 — Constructive and digital
Digital (138, 159, 17717)-net over F3, using
- 33 times duplication [i] based on digital (135, 156, 17717)-net over F3, using
- net defined by OOA [i] based on linear OOA(3156, 17717, F3, 21, 21) (dual of [(17717, 21), 371901, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3156, 177171, F3, 21) (dual of [177171, 177015, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(3155, 177148, F3, 21) (dual of [177148, 176993, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(3133, 177148, F3, 19) (dual of [177148, 177015, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(31, 23, F3, 1) (dual of [23, 22, 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([0,10]) ⊂ C([0,9]) [i] based on
- OOA 10-folding and stacking with additional row [i] based on linear OA(3156, 177171, F3, 21) (dual of [177171, 177015, 22]-code), using
- net defined by OOA [i] based on linear OOA(3156, 17717, F3, 21, 21) (dual of [(17717, 21), 371901, 22]-NRT-code), using
(138, 138+21, 59058)-Net over F3 — Digital
Digital (138, 159, 59058)-net over F3, using
- 31 times duplication [i] based on digital (137, 158, 59058)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3158, 59058, F3, 3, 21) (dual of [(59058, 3), 177016, 22]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3158, 177174, F3, 21) (dual of [177174, 177016, 22]-code), using
- 2 times code embedding in larger space [i] based on linear OA(3156, 177172, F3, 21) (dual of [177172, 177016, 22]-code), using
- construction X4 applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(3155, 177148, F3, 21) (dual of [177148, 176993, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(3133, 177148, F3, 19) (dual of [177148, 177015, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (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([0,10]) ⊂ C([0,9]) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(3156, 177172, F3, 21) (dual of [177172, 177016, 22]-code), using
- OOA 3-folding [i] based on linear OA(3158, 177174, F3, 21) (dual of [177174, 177016, 22]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3158, 59058, F3, 3, 21) (dual of [(59058, 3), 177016, 22]-NRT-code), using
(138, 138+21, large)-Net in Base 3 — Upper bound on s
There is no (138, 159, large)-net in base 3, because
- 19 times m-reduction [i] would yield (138, 140, large)-net in base 3, but