Best Known (129, 129+17, s)-Nets in Base 3
(129, 129+17, 199292)-Net over F3 — Constructive and digital
Digital (129, 146, 199292)-net over F3, using
- 31 times duplication [i] based on digital (128, 145, 199292)-net over F3, using
- net defined by OOA [i] based on linear OOA(3145, 199292, F3, 17, 17) (dual of [(199292, 17), 3387819, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3145, 1594337, F3, 17) (dual of [1594337, 1594192, 18]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3144, 1594336, F3, 17) (dual of [1594336, 1594192, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- 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(3131, 1594323, F3, 16) (dual of [1594323, 1594192, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(30, 13, F3, 0) (dual of [13, 13, 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(16) ⊂ Ce(15) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3144, 1594336, F3, 17) (dual of [1594336, 1594192, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3145, 1594337, F3, 17) (dual of [1594337, 1594192, 18]-code), using
- net defined by OOA [i] based on linear OOA(3145, 199292, F3, 17, 17) (dual of [(199292, 17), 3387819, 18]-NRT-code), using
(129, 129+17, 501995)-Net over F3 — Digital
Digital (129, 146, 501995)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3146, 501995, F3, 3, 17) (dual of [(501995, 3), 1505839, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3146, 531446, F3, 3, 17) (dual of [(531446, 3), 1594192, 18]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3146, 1594338, F3, 17) (dual of [1594338, 1594192, 18]-code), using
- 2 times code embedding in larger space [i] based on linear OA(3144, 1594336, F3, 17) (dual of [1594336, 1594192, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- 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(3131, 1594323, F3, 16) (dual of [1594323, 1594192, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(30, 13, F3, 0) (dual of [13, 13, 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(16) ⊂ Ce(15) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(3144, 1594336, F3, 17) (dual of [1594336, 1594192, 18]-code), using
- OOA 3-folding [i] based on linear OA(3146, 1594338, F3, 17) (dual of [1594338, 1594192, 18]-code), using
- discarding factors / shortening the dual code based on linear OOA(3146, 531446, F3, 3, 17) (dual of [(531446, 3), 1594192, 18]-NRT-code), using
(129, 129+17, large)-Net in Base 3 — Upper bound on s
There is no (129, 146, large)-net in base 3, because
- 15 times m-reduction [i] would yield (129, 131, large)-net in base 3, but