Best Known (39−5, 39, s)-Nets in Base 3
(39−5, 39, 265730)-Net over F3 — Constructive and digital
Digital (34, 39, 265730)-net over F3, using
- net defined by OOA [i] based on linear OOA(339, 265730, F3, 5, 5) (dual of [(265730, 5), 1328611, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(339, 265730, F3, 4, 5) (dual of [(265730, 4), 1062881, 6]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(32, 4, F3, 4, 2) (dual of [(4, 4), 14, 3]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(4;14,3) [i]
- linear OOA(337, 265726, F3, 4, 5) (dual of [(265726, 4), 1062867, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(337, 531453, F3, 5) (dual of [531453, 531416, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- 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(325, 531441, F3, 4) (dual of [531441, 531416, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [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(4) ⊂ Ce(3) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(337, 531453, F3, 5) (dual of [531453, 531416, 6]-code), using
- linear OOA(32, 4, F3, 4, 2) (dual of [(4, 4), 14, 3]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(339, 265730, F3, 4, 5) (dual of [(265730, 4), 1062881, 6]-NRT-code), using
(39−5, 39, 531457)-Net over F3 — Digital
Digital (34, 39, 531457)-net over F3, using
- net defined by OOA [i] based on linear OOA(339, 531457, F3, 5, 5) (dual of [(531457, 5), 2657246, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(339, 531457, F3, 4, 5) (dual of [(531457, 4), 2125789, 6]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(339, 531457, F3, 5) (dual of [531457, 531418, 6]-code), using
- (u, u+v)-construction [i] based on
- linear OA(32, 4, F3, 2) (dual of [4, 2, 3]-code or 4-arc in PG(1,3)), using
- extended Reed–Solomon code RSe(2,3) [i]
- Hamming code H(2,3) [i]
- Simplex code S(2,3) [i]
- the Tetracode [i]
- linear OA(337, 531453, F3, 5) (dual of [531453, 531416, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- 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(325, 531441, F3, 4) (dual of [531441, 531416, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [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(4) ⊂ Ce(3) [i] based on
- linear OA(32, 4, F3, 2) (dual of [4, 2, 3]-code or 4-arc in PG(1,3)), using
- (u, u+v)-construction [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(339, 531457, F3, 5) (dual of [531457, 531418, 6]-code), using
- appending kth column [i] based on linear OOA(339, 531457, F3, 4, 5) (dual of [(531457, 4), 2125789, 6]-NRT-code), using
(39−5, 39, large)-Net in Base 3 — Upper bound on s
There is no (34, 39, large)-net in base 3, because
- 3 times m-reduction [i] would yield (34, 36, large)-net in base 3, but