Best Known (54, 54+8, s)-Nets in Base 3
(54, 54+8, 132863)-Net over F3 — Constructive and digital
Digital (54, 62, 132863)-net over F3, using
- 31 times duplication [i] based on digital (53, 61, 132863)-net over F3, using
- net defined by OOA [i] based on linear OOA(361, 132863, F3, 8, 8) (dual of [(132863, 8), 1062843, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(361, 531452, F3, 8) (dual of [531452, 531391, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(361, 531453, F3, 8) (dual of [531453, 531392, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(361, 531441, F3, 8) (dual of [531441, 531380, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- 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(30, 12, F3, 0) (dual of [12, 12, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(361, 531453, F3, 8) (dual of [531453, 531392, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(361, 531452, F3, 8) (dual of [531452, 531391, 9]-code), using
- net defined by OOA [i] based on linear OOA(361, 132863, F3, 8, 8) (dual of [(132863, 8), 1062843, 9]-NRT-code), using
(54, 54+8, 265727)-Net over F3 — Digital
Digital (54, 62, 265727)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(362, 265727, F3, 2, 8) (dual of [(265727, 2), 531392, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(362, 531454, F3, 8) (dual of [531454, 531392, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(362, 531455, F3, 8) (dual of [531455, 531393, 9]-code), using
- construction X4 applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(361, 531441, F3, 8) (dual of [531441, 531380, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- 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(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(7) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(362, 531455, F3, 8) (dual of [531455, 531393, 9]-code), using
- OOA 2-folding [i] based on linear OA(362, 531454, F3, 8) (dual of [531454, 531392, 9]-code), using
(54, 54+8, large)-Net in Base 3 — Upper bound on s
There is no (54, 62, large)-net in base 3, because
- 6 times m-reduction [i] would yield (54, 56, large)-net in base 3, but