Best Known (139, 139+18, s)-Nets in Base 3
(139, 139+18, 177148)-Net over F3 — Constructive and digital
Digital (139, 157, 177148)-net over F3, using
- net defined by OOA [i] based on linear OOA(3157, 177148, F3, 18, 18) (dual of [(177148, 18), 3188507, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(3157, 1594332, F3, 18) (dual of [1594332, 1594175, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(3157, 1594336, F3, 18) (dual of [1594336, 1594179, 19]-code), using
- 1 times truncation [i] based on linear OA(3158, 1594337, F3, 19) (dual of [1594337, 1594179, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- linear OA(3157, 1594323, F3, 19) (dual of [1594323, 1594166, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(3144, 1594323, F3, 17) (dual of [1594323, 1594179, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(31, 14, F3, 1) (dual of [14, 13, 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(18) ⊂ Ce(16) [i] based on
- 1 times truncation [i] based on linear OA(3158, 1594337, F3, 19) (dual of [1594337, 1594179, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3157, 1594336, F3, 18) (dual of [1594336, 1594179, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(3157, 1594332, F3, 18) (dual of [1594332, 1594175, 19]-code), using
(139, 139+18, 531445)-Net over F3 — Digital
Digital (139, 157, 531445)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3157, 531445, F3, 3, 18) (dual of [(531445, 3), 1594178, 19]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3157, 1594335, F3, 18) (dual of [1594335, 1594178, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(3157, 1594336, F3, 18) (dual of [1594336, 1594179, 19]-code), using
- 1 times truncation [i] based on linear OA(3158, 1594337, F3, 19) (dual of [1594337, 1594179, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- linear OA(3157, 1594323, F3, 19) (dual of [1594323, 1594166, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(3144, 1594323, F3, 17) (dual of [1594323, 1594179, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(31, 14, F3, 1) (dual of [14, 13, 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(18) ⊂ Ce(16) [i] based on
- 1 times truncation [i] based on linear OA(3158, 1594337, F3, 19) (dual of [1594337, 1594179, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3157, 1594336, F3, 18) (dual of [1594336, 1594179, 19]-code), using
- OOA 3-folding [i] based on linear OA(3157, 1594335, F3, 18) (dual of [1594335, 1594178, 19]-code), using
(139, 139+18, large)-Net in Base 3 — Upper bound on s
There is no (139, 157, large)-net in base 3, because
- 16 times m-reduction [i] would yield (139, 141, large)-net in base 3, but