Best Known (162−27, 162, s)-Nets in Base 3
(162−27, 162, 1513)-Net over F3 — Constructive and digital
Digital (135, 162, 1513)-net over F3, using
- net defined by OOA [i] based on linear OOA(3162, 1513, F3, 27, 27) (dual of [(1513, 27), 40689, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3162, 19670, F3, 27) (dual of [19670, 19508, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3162, 19682, F3, 27) (dual of [19682, 19520, 28]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(3163, 19683, F3, 28) (dual of [19683, 19520, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3162, 19682, F3, 27) (dual of [19682, 19520, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3162, 19670, F3, 27) (dual of [19670, 19508, 28]-code), using
(162−27, 162, 7410)-Net over F3 — Digital
Digital (135, 162, 7410)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3162, 7410, F3, 2, 27) (dual of [(7410, 2), 14658, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3162, 9841, F3, 2, 27) (dual of [(9841, 2), 19520, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3162, 19682, F3, 27) (dual of [19682, 19520, 28]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(3163, 19683, F3, 28) (dual of [19683, 19520, 29]-code), using
- OOA 2-folding [i] based on linear OA(3162, 19682, F3, 27) (dual of [19682, 19520, 28]-code), using
- discarding factors / shortening the dual code based on linear OOA(3162, 9841, F3, 2, 27) (dual of [(9841, 2), 19520, 28]-NRT-code), using
(162−27, 162, 2297931)-Net in Base 3 — Upper bound on s
There is no (135, 162, 2297932)-net in base 3, because
- 1 times m-reduction [i] would yield (135, 161, 2297932)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 65542 588212 041740 990529 264936 372200 380783 703475 182168 539526 728780 018583 608441 > 3161 [i]