Best Known (53, 53+10, s)-Nets in Base 3
(53, 53+10, 11812)-Net over F3 — Constructive and digital
Digital (53, 63, 11812)-net over F3, using
- 31 times duplication [i] based on digital (52, 62, 11812)-net over F3, using
- net defined by OOA [i] based on linear OOA(362, 11812, F3, 10, 10) (dual of [(11812, 10), 118058, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(362, 59060, F3, 10) (dual of [59060, 58998, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(361, 59049, F3, 10) (dual of [59049, 58988, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(351, 59049, F3, 8) (dual of [59049, 58998, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(31, 11, F3, 1) (dual of [11, 10, 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
- OA 5-folding and stacking [i] based on linear OA(362, 59060, F3, 10) (dual of [59060, 58998, 11]-code), using
- net defined by OOA [i] based on linear OOA(362, 11812, F3, 10, 10) (dual of [(11812, 10), 118058, 11]-NRT-code), using
(53, 53+10, 24297)-Net over F3 — Digital
Digital (53, 63, 24297)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(363, 24297, F3, 2, 10) (dual of [(24297, 2), 48531, 11]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(363, 29531, F3, 2, 10) (dual of [(29531, 2), 58999, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(363, 59062, F3, 10) (dual of [59062, 58999, 11]-code), using
- 1 times code embedding in larger space [i] based on linear OA(362, 59061, F3, 10) (dual of [59061, 58999, 11]-code), using
- construction X4 applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(361, 59049, F3, 10) (dual of [59049, 58988, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(351, 59049, F3, 8) (dual of [59049, 58998, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(311, 12, F3, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,3)), using
- dual of repetition code with length 12 [i]
- linear OA(31, 12, F3, 1) (dual of [12, 11, 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
- 1 times code embedding in larger space [i] based on linear OA(362, 59061, F3, 10) (dual of [59061, 58999, 11]-code), using
- OOA 2-folding [i] based on linear OA(363, 59062, F3, 10) (dual of [59062, 58999, 11]-code), using
- discarding factors / shortening the dual code based on linear OOA(363, 29531, F3, 2, 10) (dual of [(29531, 2), 58999, 11]-NRT-code), using
(53, 53+10, 1338235)-Net in Base 3 — Upper bound on s
There is no (53, 63, 1338236)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 1 144562 670872 039072 797447 892697 > 363 [i]