Best Known (19−5, 19, s)-Nets in Base 3
(19−5, 19, 367)-Net over F3 — Constructive and digital
Digital (14, 19, 367)-net over F3, using
- net defined by OOA [i] based on linear OOA(319, 367, F3, 5, 5) (dual of [(367, 5), 1816, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(319, 735, F3, 5) (dual of [735, 716, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(319, 729, F3, 5) (dual of [729, 710, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(313, 729, F3, 4) (dual of [729, 716, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(30, 6, F3, 0) (dual of [6, 6, 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(319, 735, F3, 5) (dual of [735, 716, 6]-code), using
(19−5, 19, 660)-Net over F3 — Digital
Digital (14, 19, 660)-net over F3, using
- net defined by OOA [i] based on linear OOA(319, 660, F3, 5, 5) (dual of [(660, 5), 3281, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(319, 660, F3, 4, 5) (dual of [(660, 4), 2621, 6]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(319, 660, F3, 5) (dual of [660, 641, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(319, 728, F3, 5) (dual of [728, 709, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,4], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(319, 728, F3, 5) (dual of [728, 709, 6]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(319, 660, F3, 5) (dual of [660, 641, 6]-code), using
- appending kth column [i] based on linear OOA(319, 660, F3, 4, 5) (dual of [(660, 4), 2621, 6]-NRT-code), using
(19−5, 19, 13916)-Net in Base 3 — Upper bound on s
There is no (14, 19, 13917)-net in base 3, because
- 1 times m-reduction [i] would yield (14, 18, 13917)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 387 449281 > 318 [i]