Best Known (146−25, 146, s)-Nets in Base 3
(146−25, 146, 1641)-Net over F3 — Constructive and digital
Digital (121, 146, 1641)-net over F3, using
- net defined by OOA [i] based on linear OOA(3146, 1641, F3, 25, 25) (dual of [(1641, 25), 40879, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3146, 19693, F3, 25) (dual of [19693, 19547, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- linear OA(3145, 19683, F3, 25) (dual of [19683, 19538, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3136, 19683, F3, 23) (dual of [19683, 19547, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(24) ⊂ Ce(22) [i] based on
- OOA 12-folding and stacking with additional row [i] based on linear OA(3146, 19693, F3, 25) (dual of [19693, 19547, 26]-code), using
(146−25, 146, 6564)-Net over F3 — Digital
Digital (121, 146, 6564)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3146, 6564, F3, 3, 25) (dual of [(6564, 3), 19546, 26]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3146, 19692, F3, 25) (dual of [19692, 19546, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3146, 19693, F3, 25) (dual of [19693, 19547, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- linear OA(3145, 19683, F3, 25) (dual of [19683, 19538, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3136, 19683, F3, 23) (dual of [19683, 19547, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(3146, 19693, F3, 25) (dual of [19693, 19547, 26]-code), using
- OOA 3-folding [i] based on linear OA(3146, 19692, F3, 25) (dual of [19692, 19546, 26]-code), using
(146−25, 146, 1540080)-Net in Base 3 — Upper bound on s
There is no (121, 146, 1540081)-net in base 3, because
- 1 times m-reduction [i] would yield (121, 145, 1540081)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1522 595895 913913 773885 215969 505857 381227 164686 407300 706373 090187 389809 > 3145 [i]