Best Known (53−8, 53, s)-Nets in Base 3
(53−8, 53, 14765)-Net over F3 — Constructive and digital
Digital (45, 53, 14765)-net over F3, using
- 31 times duplication [i] based on digital (44, 52, 14765)-net over F3, using
- net defined by OOA [i] based on linear OOA(352, 14765, F3, 8, 8) (dual of [(14765, 8), 118068, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(352, 59060, F3, 8) (dual of [59060, 59008, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(352, 59061, F3, 8) (dual of [59061, 59009, 9]-code), using
- construction X4 applied to Ce(7) ⊂ Ce(6) [i] based on
- 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(341, 59049, F3, 7) (dual of [59049, 59008, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(7) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(352, 59061, F3, 8) (dual of [59061, 59009, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(352, 59060, F3, 8) (dual of [59060, 59008, 9]-code), using
- net defined by OOA [i] based on linear OOA(352, 14765, F3, 8, 8) (dual of [(14765, 8), 118068, 9]-NRT-code), using
(53−8, 53, 29531)-Net over F3 — Digital
Digital (45, 53, 29531)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(353, 29531, F3, 2, 8) (dual of [(29531, 2), 59009, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(353, 59062, F3, 8) (dual of [59062, 59009, 9]-code), using
- 1 times code embedding in larger space [i] based on linear OA(352, 59061, F3, 8) (dual of [59061, 59009, 9]-code), using
- construction X4 applied to Ce(7) ⊂ Ce(6) [i] based on
- 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(341, 59049, F3, 7) (dual of [59049, 59008, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(7) ⊂ Ce(6) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(352, 59061, F3, 8) (dual of [59061, 59009, 9]-code), using
- OOA 2-folding [i] based on linear OA(353, 59062, F3, 8) (dual of [59062, 59009, 9]-code), using
(53−8, 53, 2322088)-Net in Base 3 — Upper bound on s
There is no (45, 53, 2322089)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 19 383259 229293 433444 802833 > 353 [i]