Best Known (33−5, 33, s)-Nets in Base 3
(33−5, 33, 29533)-Net over F3 — Constructive and digital
Digital (28, 33, 29533)-net over F3, using
- net defined by OOA [i] based on linear OOA(333, 29533, F3, 5, 5) (dual of [(29533, 5), 147632, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(333, 29533, F3, 4, 5) (dual of [(29533, 4), 118099, 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(331, 29529, F3, 4, 5) (dual of [(29529, 4), 118085, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(331, 59059, F3, 5) (dual of [59059, 59028, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(331, 59049, F3, 5) (dual of [59049, 59018, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(321, 59049, F3, 4) (dual of [59049, 59028, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(30, 10, F3, 0) (dual of [10, 10, 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(331, 59059, F3, 5) (dual of [59059, 59028, 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(333, 29533, F3, 4, 5) (dual of [(29533, 4), 118099, 6]-NRT-code), using
(33−5, 33, 59063)-Net over F3 — Digital
Digital (28, 33, 59063)-net over F3, using
- net defined by OOA [i] based on linear OOA(333, 59063, F3, 5, 5) (dual of [(59063, 5), 295282, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(333, 59063, F3, 4, 5) (dual of [(59063, 4), 236219, 6]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(333, 59063, F3, 5) (dual of [59063, 59030, 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(331, 59059, F3, 5) (dual of [59059, 59028, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(331, 59049, F3, 5) (dual of [59049, 59018, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(321, 59049, F3, 4) (dual of [59049, 59028, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(30, 10, F3, 0) (dual of [10, 10, 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(333, 59063, F3, 5) (dual of [59063, 59030, 6]-code), using
- appending kth column [i] based on linear OOA(333, 59063, F3, 4, 5) (dual of [(59063, 4), 236219, 6]-NRT-code), using
(33−5, 33, large)-Net in Base 3 — Upper bound on s
There is no (28, 33, large)-net in base 3, because
- 3 times m-reduction [i] would yield (28, 30, large)-net in base 3, but