Best Known (153−18, 153, s)-Nets in Base 3
(153−18, 153, 59053)-Net over F3 — Constructive and digital
Digital (135, 153, 59053)-net over F3, using
- t-expansion [i] based on digital (134, 153, 59053)-net over F3, using
- net defined by OOA [i] based on linear OOA(3153, 59053, F3, 19, 19) (dual of [(59053, 19), 1121854, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3153, 531478, F3, 19) (dual of [531478, 531325, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3153, 531482, F3, 19) (dual of [531482, 531329, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(3145, 531441, F3, 19) (dual of [531441, 531296, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(3109, 531441, F3, 14) (dual of [531441, 531332, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(18) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(3153, 531482, F3, 19) (dual of [531482, 531329, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3153, 531478, F3, 19) (dual of [531478, 531325, 20]-code), using
- net defined by OOA [i] based on linear OOA(3153, 59053, F3, 19, 19) (dual of [(59053, 19), 1121854, 20]-NRT-code), using
(153−18, 153, 204046)-Net over F3 — Digital
Digital (135, 153, 204046)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3153, 204046, F3, 2, 18) (dual of [(204046, 2), 407939, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3153, 265742, F3, 2, 18) (dual of [(265742, 2), 531331, 19]-NRT-code), using
- 1 step truncation [i] based on linear OOA(3154, 265743, F3, 2, 19) (dual of [(265743, 2), 531332, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3154, 531486, F3, 19) (dual of [531486, 531332, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(3145, 531441, F3, 19) (dual of [531441, 531296, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(3109, 531441, F3, 14) (dual of [531441, 531332, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(39, 45, F3, 4) (dual of [45, 36, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- OOA 2-folding [i] based on linear OA(3154, 531486, F3, 19) (dual of [531486, 531332, 20]-code), using
- 1 step truncation [i] based on linear OOA(3154, 265743, F3, 2, 19) (dual of [(265743, 2), 531332, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3153, 265742, F3, 2, 18) (dual of [(265742, 2), 531331, 19]-NRT-code), using
(153−18, 153, large)-Net in Base 3 — Upper bound on s
There is no (135, 153, large)-net in base 3, because
- 16 times m-reduction [i] would yield (135, 137, large)-net in base 3, but