Best Known (144, 171, s)-Nets in Base 3
(144, 171, 1517)-Net over F3 — Constructive and digital
Digital (144, 171, 1517)-net over F3, using
- net defined by OOA [i] based on linear OOA(3171, 1517, F3, 27, 27) (dual of [(1517, 27), 40788, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3171, 19722, F3, 27) (dual of [19722, 19551, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3171, 19724, F3, 27) (dual of [19724, 19553, 28]-code), using
- construction X applied to Ce(27) ⊂ Ce(21) [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(3127, 19683, F3, 22) (dual of [19683, 19556, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- construction X applied to Ce(27) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3171, 19724, F3, 27) (dual of [19724, 19553, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3171, 19722, F3, 27) (dual of [19722, 19551, 28]-code), using
(144, 171, 9862)-Net over F3 — Digital
Digital (144, 171, 9862)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3171, 9862, F3, 2, 27) (dual of [(9862, 2), 19553, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3171, 19724, F3, 27) (dual of [19724, 19553, 28]-code), using
- construction X applied to Ce(27) ⊂ Ce(21) [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(3127, 19683, F3, 22) (dual of [19683, 19556, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- construction X applied to Ce(27) ⊂ Ce(21) [i] based on
- OOA 2-folding [i] based on linear OA(3171, 19724, F3, 27) (dual of [19724, 19553, 28]-code), using
(144, 171, 4916465)-Net in Base 3 — Upper bound on s
There is no (144, 171, 4916466)-net in base 3, because
- 1 times m-reduction [i] would yield (144, 170, 4916466)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1290 070675 038562 763507 011027 714574 162334 191964 349247 587214 430539 150056 863849 842349 > 3170 [i]