Best Known (170−27, 170, s)-Nets in Base 3
(170−27, 170, 1516)-Net over F3 — Constructive and digital
Digital (143, 170, 1516)-net over F3, using
- 32 times duplication [i] based on digital (141, 168, 1516)-net over F3, using
- net defined by OOA [i] based on linear OOA(3168, 1516, F3, 27, 27) (dual of [(1516, 27), 40764, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3168, 19709, F3, 27) (dual of [19709, 19541, 28]-code), using
- construction XX applied to Ce(27) ⊂ Ce(24) ⊂ Ce(22) [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(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, 22, F3, 1) (dual of [22, 21, 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
- linear OA(31, 4, F3, 1) (dual of [4, 3, 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 (see above)
- construction XX applied to Ce(27) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- OOA 13-folding and stacking with additional row [i] based on linear OA(3168, 19709, F3, 27) (dual of [19709, 19541, 28]-code), using
- net defined by OOA [i] based on linear OOA(3168, 1516, F3, 27, 27) (dual of [(1516, 27), 40764, 28]-NRT-code), using
(170−27, 170, 9858)-Net over F3 — Digital
Digital (143, 170, 9858)-net over F3, using
- 31 times duplication [i] based on digital (142, 169, 9858)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3169, 9858, F3, 2, 27) (dual of [(9858, 2), 19547, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3169, 19716, F3, 27) (dual of [19716, 19547, 28]-code), using
- construction X applied to Ce(27) ⊂ Ce(22) [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(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(36, 33, F3, 3) (dual of [33, 27, 4]-code or 33-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(27) ⊂ Ce(22) [i] based on
- OOA 2-folding [i] based on linear OA(3169, 19716, F3, 27) (dual of [19716, 19547, 28]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3169, 9858, F3, 2, 27) (dual of [(9858, 2), 19547, 28]-NRT-code), using
(170−27, 170, 4518052)-Net in Base 3 — Upper bound on s
There is no (143, 170, 4518053)-net in base 3, because
- 1 times m-reduction [i] would yield (143, 169, 4518053)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 430 023731 066867 539822 796584 415653 222601 086609 877020 655717 453040 572353 253507 599419 > 3169 [i]