Best Known (52−10, 52, s)-Nets in Base 3
(52−10, 52, 1314)-Net over F3 — Constructive and digital
Digital (42, 52, 1314)-net over F3, using
- 32 times duplication [i] based on digital (40, 50, 1314)-net over F3, using
- net defined by OOA [i] based on linear OOA(350, 1314, F3, 10, 10) (dual of [(1314, 10), 13090, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(350, 6570, F3, 10) (dual of [6570, 6520, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(349, 6561, F3, 10) (dual of [6561, 6512, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(341, 6561, F3, 8) (dual of [6561, 6520, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(31, 9, F3, 1) (dual of [9, 8, 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(350, 6570, F3, 10) (dual of [6570, 6520, 11]-code), using
- net defined by OOA [i] based on linear OOA(350, 1314, F3, 10, 10) (dual of [(1314, 10), 13090, 11]-NRT-code), using
(52−10, 52, 3287)-Net over F3 — Digital
Digital (42, 52, 3287)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(352, 3287, F3, 2, 10) (dual of [(3287, 2), 6522, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(352, 6574, F3, 10) (dual of [6574, 6522, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- linear OA(349, 6561, F3, 10) (dual of [6561, 6512, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(333, 6561, F3, 7) (dual of [6561, 6528, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- OOA 2-folding [i] based on linear OA(352, 6574, F3, 10) (dual of [6574, 6522, 11]-code), using
(52−10, 52, 119357)-Net in Base 3 — Upper bound on s
There is no (42, 52, 119358)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 6 461126 778300 237931 706005 > 352 [i]