Best Known (137−15, 137, s)-Nets in Base 3
(137−15, 137, 227766)-Net over F3 — Constructive and digital
Digital (122, 137, 227766)-net over F3, using
- net defined by OOA [i] based on linear OOA(3137, 227766, F3, 15, 15) (dual of [(227766, 15), 3416353, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3137, 1594363, F3, 15) (dual of [1594363, 1594226, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(3137, 1594368, F3, 15) (dual of [1594368, 1594231, 16]-code), using
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- 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(392, 1594323, F3, 11) (dual of [1594323, 1594231, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(36, 45, F3, 3) (dual of [45, 39, 4]-code or 45-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(3137, 1594368, F3, 15) (dual of [1594368, 1594231, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3137, 1594363, F3, 15) (dual of [1594363, 1594226, 16]-code), using
(137−15, 137, 616507)-Net over F3 — Digital
Digital (122, 137, 616507)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3137, 616507, F3, 2, 15) (dual of [(616507, 2), 1232877, 16]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3137, 797184, F3, 2, 15) (dual of [(797184, 2), 1594231, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3137, 1594368, F3, 15) (dual of [1594368, 1594231, 16]-code), using
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- 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(392, 1594323, F3, 11) (dual of [1594323, 1594231, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(36, 45, F3, 3) (dual of [45, 39, 4]-code or 45-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- OOA 2-folding [i] based on linear OA(3137, 1594368, F3, 15) (dual of [1594368, 1594231, 16]-code), using
- discarding factors / shortening the dual code based on linear OOA(3137, 797184, F3, 2, 15) (dual of [(797184, 2), 1594231, 16]-NRT-code), using
(137−15, 137, large)-Net in Base 3 — Upper bound on s
There is no (122, 137, large)-net in base 3, because
- 13 times m-reduction [i] would yield (122, 124, large)-net in base 3, but