Best Known (42−7, 42, s)-Nets in Base 3
(42−7, 42, 19686)-Net over F3 — Constructive and digital
Digital (35, 42, 19686)-net over F3, using
- net defined by OOA [i] based on linear OOA(342, 19686, F3, 7, 7) (dual of [(19686, 7), 137760, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(342, 59059, F3, 7) (dual of [59059, 59017, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(342, 59060, F3, 7) (dual of [59060, 59018, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(4) [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(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(31, 11, F3, 1) (dual of [11, 10, 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(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(342, 59060, F3, 7) (dual of [59060, 59018, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(342, 59059, F3, 7) (dual of [59059, 59017, 8]-code), using
(42−7, 42, 29530)-Net over F3 — Digital
Digital (35, 42, 29530)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(342, 29530, F3, 2, 7) (dual of [(29530, 2), 59018, 8]-NRT-code), using
- OOA 2-folding [i] based on linear OA(342, 59060, F3, 7) (dual of [59060, 59018, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(4) [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(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(31, 11, F3, 1) (dual of [11, 10, 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(4) [i] based on
- OOA 2-folding [i] based on linear OA(342, 59060, F3, 7) (dual of [59060, 59018, 8]-code), using
(42−7, 42, 3013079)-Net in Base 3 — Upper bound on s
There is no (35, 42, 3013080)-net in base 3, because
- 1 times m-reduction [i] would yield (35, 41, 3013080)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 36 473026 807101 871841 > 341 [i]