Best Known (36−5, 36, s)-Nets in Base 3
(36−5, 36, 88582)-Net over F3 — Constructive and digital
Digital (31, 36, 88582)-net over F3, using
- net defined by OOA [i] based on linear OOA(336, 88582, F3, 5, 5) (dual of [(88582, 5), 442874, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(336, 88583, F3, 2, 5) (dual of [(88583, 2), 177130, 6]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(32, 4, F3, 2, 2) (dual of [(4, 2), 6, 3]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(2;6,3) [i]
- linear OOA(334, 88579, F3, 2, 5) (dual of [(88579, 2), 177124, 6]-NRT-code), using
- OOA 2-folding [i] based on linear OA(334, 177158, F3, 5) (dual of [177158, 177124, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(334, 177147, F3, 5) (dual of [177147, 177113, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(323, 177147, F3, 4) (dual of [177147, 177124, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(30, 11, F3, 0) (dual of [11, 11, 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 [i] based on linear OA(334, 177158, F3, 5) (dual of [177158, 177124, 6]-code), using
- linear OOA(32, 4, F3, 2, 2) (dual of [(4, 2), 6, 3]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(336, 88583, F3, 2, 5) (dual of [(88583, 2), 177130, 6]-NRT-code), using
(36−5, 36, 177162)-Net over F3 — Digital
Digital (31, 36, 177162)-net over F3, using
- net defined by OOA [i] based on linear OOA(336, 177162, F3, 5, 5) (dual of [(177162, 5), 885774, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(336, 177162, F3, 4, 5) (dual of [(177162, 4), 708612, 6]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(336, 177162, F3, 5) (dual of [177162, 177126, 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(334, 177158, F3, 5) (dual of [177158, 177124, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(334, 177147, F3, 5) (dual of [177147, 177113, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(323, 177147, F3, 4) (dual of [177147, 177124, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(30, 11, F3, 0) (dual of [11, 11, 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(336, 177162, F3, 5) (dual of [177162, 177126, 6]-code), using
- appending kth column [i] based on linear OOA(336, 177162, F3, 4, 5) (dual of [(177162, 4), 708612, 6]-NRT-code), using
(36−5, 36, large)-Net in Base 3 — Upper bound on s
There is no (31, 36, large)-net in base 3, because
- 3 times m-reduction [i] would yield (31, 33, large)-net in base 3, but