Best Known (37, 37+6, s)-Nets in Base 3
(37, 37+6, 19690)-Net over F3 — Constructive and digital
Digital (37, 43, 19690)-net over F3, using
- 31 times duplication [i] based on digital (36, 42, 19690)-net over F3, using
- net defined by OOA [i] based on linear OOA(342, 19690, F3, 6, 6) (dual of [(19690, 6), 118098, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(342, 19690, F3, 5, 6) (dual of [(19690, 5), 98408, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(342, 59070, F3, 6) (dual of [59070, 59028, 7]-code), using
- construction X applied to Ce(6) ⊂ Ce(3) [i] based on
- linear OA(341, 59049, F3, 7) (dual of [59049, 59008, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(31, 21, F3, 1) (dual of [21, 20, 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 X applied to Ce(6) ⊂ Ce(3) [i] based on
- OA 3-folding and stacking [i] based on linear OA(342, 59070, F3, 6) (dual of [59070, 59028, 7]-code), using
- appending kth column [i] based on linear OOA(342, 19690, F3, 5, 6) (dual of [(19690, 5), 98408, 7]-NRT-code), using
- net defined by OOA [i] based on linear OOA(342, 19690, F3, 6, 6) (dual of [(19690, 6), 118098, 7]-NRT-code), using
(37, 37+6, 59073)-Net over F3 — Digital
Digital (37, 43, 59073)-net over F3, using
- net defined by OOA [i] based on linear OOA(343, 59073, F3, 6, 6) (dual of [(59073, 6), 354395, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(343, 59073, F3, 5, 6) (dual of [(59073, 5), 295322, 7]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(343, 59073, F3, 6) (dual of [59073, 59030, 7]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(342, 59071, F3, 6) (dual of [59071, 59029, 7]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(3) [i] based on
- linear OA(341, 59049, F3, 7) (dual of [59049, 59008, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(321, 22, F3, 21) (dual of [22, 1, 22]-code or 22-arc in PG(20,3)), using
- dual of repetition code with length 22 [i]
- linear OA(31, 22, F3, 1) (dual of [22, 21, 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(3) [i] based on
- linear OA(342, 59072, F3, 5) (dual of [59072, 59030, 6]-code), using Gilbert–Varšamov bound and bm = 342 > Vbs−1(k−1) = 8 116646 366233 929483 [i]
- linear OA(30, 1, F3, 0) (dual of [1, 1, 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
- linear OA(342, 59071, F3, 6) (dual of [59071, 59029, 7]-code), using
- construction X with Varšamov bound [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(343, 59073, F3, 6) (dual of [59073, 59030, 7]-code), using
- appending kth column [i] based on linear OOA(343, 59073, F3, 5, 6) (dual of [(59073, 5), 295322, 7]-NRT-code), using
(37, 37+6, 6267459)-Net in Base 3 — Upper bound on s
There is no (37, 43, 6267460)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 328 256979 530919 020081 > 343 [i]