Best Known (117, 117+17, s)-Nets in Base 3
(117, 117+17, 66431)-Net over F3 — Constructive and digital
Digital (117, 134, 66431)-net over F3, using
- 31 times duplication [i] based on digital (116, 133, 66431)-net over F3, using
- net defined by OOA [i] based on linear OOA(3133, 66431, F3, 17, 17) (dual of [(66431, 17), 1129194, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3133, 531449, F3, 17) (dual of [531449, 531316, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(3133, 531453, F3, 17) (dual of [531453, 531320, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(3133, 531441, F3, 17) (dual of [531441, 531308, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(3121, 531441, F3, 16) (dual of [531441, 531320, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(30, 12, F3, 0) (dual of [12, 12, 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
- discarding factors / shortening the dual code based on linear OA(3133, 531453, F3, 17) (dual of [531453, 531320, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3133, 531449, F3, 17) (dual of [531449, 531316, 18]-code), using
- net defined by OOA [i] based on linear OOA(3133, 66431, F3, 17, 17) (dual of [(66431, 17), 1129194, 18]-NRT-code), using
(117, 117+17, 177151)-Net over F3 — Digital
Digital (117, 134, 177151)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3134, 177151, F3, 3, 17) (dual of [(177151, 3), 531319, 18]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3133, 177151, F3, 3, 17) (dual of [(177151, 3), 531320, 18]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3133, 531453, F3, 17) (dual of [531453, 531320, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(3133, 531441, F3, 17) (dual of [531441, 531308, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(3121, 531441, F3, 16) (dual of [531441, 531320, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(30, 12, F3, 0) (dual of [12, 12, 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
- OOA 3-folding [i] based on linear OA(3133, 531453, F3, 17) (dual of [531453, 531320, 18]-code), using
- 31 times duplication [i] based on linear OOA(3133, 177151, F3, 3, 17) (dual of [(177151, 3), 531320, 18]-NRT-code), using
(117, 117+17, large)-Net in Base 3 — Upper bound on s
There is no (117, 134, large)-net in base 3, because
- 15 times m-reduction [i] would yield (117, 119, large)-net in base 3, but