Best Known (165−28, 165, s)-Nets in Base 3
(165−28, 165, 1406)-Net over F3 — Constructive and digital
Digital (137, 165, 1406)-net over F3, using
- 31 times duplication [i] based on digital (136, 164, 1406)-net over F3, using
- net defined by OOA [i] based on linear OOA(3164, 1406, F3, 28, 28) (dual of [(1406, 28), 39204, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(3164, 19684, F3, 28) (dual of [19684, 19520, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3164, 19693, F3, 28) (dual of [19693, 19529, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(25) [i] based on
- linear OA(3163, 19683, F3, 28) (dual of [19683, 19520, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(3154, 19683, F3, 26) (dual of [19683, 19529, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(31, 10, F3, 1) (dual of [10, 9, 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(27) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(3164, 19693, F3, 28) (dual of [19693, 19529, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(3164, 19684, F3, 28) (dual of [19684, 19520, 29]-code), using
- net defined by OOA [i] based on linear OOA(3164, 1406, F3, 28, 28) (dual of [(1406, 28), 39204, 29]-NRT-code), using
(165−28, 165, 6564)-Net over F3 — Digital
Digital (137, 165, 6564)-net over F3, using
- 31 times duplication [i] based on digital (136, 164, 6564)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3164, 6564, F3, 3, 28) (dual of [(6564, 3), 19528, 29]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3164, 19692, F3, 28) (dual of [19692, 19528, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3164, 19693, F3, 28) (dual of [19693, 19529, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(25) [i] based on
- linear OA(3163, 19683, F3, 28) (dual of [19683, 19520, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(3154, 19683, F3, 26) (dual of [19683, 19529, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(31, 10, F3, 1) (dual of [10, 9, 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(27) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(3164, 19693, F3, 28) (dual of [19693, 19529, 29]-code), using
- OOA 3-folding [i] based on linear OA(3164, 19692, F3, 28) (dual of [19692, 19528, 29]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3164, 6564, F3, 3, 28) (dual of [(6564, 3), 19528, 29]-NRT-code), using
(165−28, 165, 1269515)-Net in Base 3 — Upper bound on s
There is no (137, 165, 1269516)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 5 308975 792222 877097 828570 454031 211575 343331 354936 524462 894871 979708 267770 055817 > 3165 [i]