Best Known (39, 50, s)-Nets in Base 3
(39, 50, 438)-Net over F3 — Constructive and digital
Digital (39, 50, 438)-net over F3, using
- net defined by OOA [i] based on linear OOA(350, 438, F3, 11, 11) (dual of [(438, 11), 4768, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(350, 2191, F3, 11) (dual of [2191, 2141, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(350, 2194, F3, 11) (dual of [2194, 2144, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(350, 2187, F3, 11) (dual of [2187, 2137, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(343, 2187, F3, 10) (dual of [2187, 2144, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(30, 7, F3, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(350, 2194, F3, 11) (dual of [2194, 2144, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(350, 2191, F3, 11) (dual of [2191, 2141, 12]-code), using
(39, 50, 1097)-Net over F3 — Digital
Digital (39, 50, 1097)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(350, 1097, F3, 2, 11) (dual of [(1097, 2), 2144, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(350, 2194, F3, 11) (dual of [2194, 2144, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(350, 2187, F3, 11) (dual of [2187, 2137, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(343, 2187, F3, 10) (dual of [2187, 2144, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(30, 7, F3, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- OOA 2-folding [i] based on linear OA(350, 2194, F3, 11) (dual of [2194, 2144, 12]-code), using
(39, 50, 61739)-Net in Base 3 — Upper bound on s
There is no (39, 50, 61740)-net in base 3, because
- 1 times m-reduction [i] would yield (39, 49, 61740)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 239309 598412 133430 388537 > 349 [i]