Best Known (50, 61, s)-Nets in Base 3
(50, 61, 1316)-Net over F3 — Constructive and digital
Digital (50, 61, 1316)-net over F3, using
- net defined by OOA [i] based on linear OOA(361, 1316, F3, 11, 11) (dual of [(1316, 11), 14415, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(361, 6581, F3, 11) (dual of [6581, 6520, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(357, 6561, F3, 11) (dual of [6561, 6504, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(34, 20, F3, 2) (dual of [20, 16, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- OOA 5-folding and stacking with additional row [i] based on linear OA(361, 6581, F3, 11) (dual of [6581, 6520, 12]-code), using
(50, 61, 3290)-Net over F3 — Digital
Digital (50, 61, 3290)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(361, 3290, F3, 2, 11) (dual of [(3290, 2), 6519, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(361, 6580, F3, 11) (dual of [6580, 6519, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(361, 6581, F3, 11) (dual of [6581, 6520, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(357, 6561, F3, 11) (dual of [6561, 6504, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(34, 20, F3, 2) (dual of [20, 16, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(361, 6581, F3, 11) (dual of [6581, 6520, 12]-code), using
- OOA 2-folding [i] based on linear OA(361, 6580, F3, 11) (dual of [6580, 6519, 12]-code), using
(50, 61, 692242)-Net in Base 3 — Upper bound on s
There is no (50, 61, 692243)-net in base 3, because
- 1 times m-reduction [i] would yield (50, 60, 692243)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 42391 200154 120740 814633 875999 > 360 [i]