Best Known (45, 45+7, s)-Nets in Base 3
(45, 45+7, 177152)-Net over F3 — Constructive and digital
Digital (45, 52, 177152)-net over F3, using
- net defined by OOA [i] based on linear OOA(352, 177152, F3, 7, 7) (dual of [(177152, 7), 1240012, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(352, 531457, F3, 7) (dual of [531457, 531405, 8]-code), using
- 2 times code embedding in larger space [i] based on linear OA(350, 531455, F3, 7) (dual of [531455, 531405, 8]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(349, 531441, F3, 7) (dual of [531441, 531392, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(337, 531441, F3, 5) (dual of [531441, 531404, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(313, 14, F3, 13) (dual of [14, 1, 14]-code or 14-arc in PG(12,3)), using
- dual of repetition code with length 14 [i]
- linear OA(31, 14, F3, 1) (dual of [14, 13, 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 X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(350, 531455, F3, 7) (dual of [531455, 531405, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(352, 531457, F3, 7) (dual of [531457, 531405, 8]-code), using
(45, 45+7, 265729)-Net over F3 — Digital
Digital (45, 52, 265729)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(352, 265729, F3, 2, 7) (dual of [(265729, 2), 531406, 8]-NRT-code), using
- OOA 2-folding [i] based on linear OA(352, 531458, F3, 7) (dual of [531458, 531406, 8]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(350, 531455, F3, 7) (dual of [531455, 531405, 8]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(349, 531441, F3, 7) (dual of [531441, 531392, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(337, 531441, F3, 5) (dual of [531441, 531404, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(313, 14, F3, 13) (dual of [14, 1, 14]-code or 14-arc in PG(12,3)), using
- dual of repetition code with length 14 [i]
- linear OA(31, 14, F3, 1) (dual of [14, 13, 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 X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(350, 531456, F3, 5) (dual of [531456, 531406, 6]-code), using Gilbert–Varšamov bound and bm = 350 > Vbs−1(k−1) = 53182 832189 907376 025611 [i]
- linear OA(31, 2, F3, 1) (dual of [2, 1, 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)
- linear OA(350, 531455, F3, 7) (dual of [531455, 531405, 8]-code), using
- construction X with Varšamov bound [i] based on
- OOA 2-folding [i] based on linear OA(352, 531458, F3, 7) (dual of [531458, 531406, 8]-code), using
(45, 45+7, large)-Net in Base 3 — Upper bound on s
There is no (45, 52, large)-net in base 3, because
- 5 times m-reduction [i] would yield (45, 47, large)-net in base 3, but