Best Known (41−6, 41, s)-Nets in Base 3
(41−6, 41, 19686)-Net over F3 — Constructive and digital
Digital (35, 41, 19686)-net over F3, using
- net defined by OOA [i] based on linear OOA(341, 19686, F3, 6, 6) (dual of [(19686, 6), 118075, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(341, 19686, F3, 5, 6) (dual of [(19686, 5), 98389, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(341, 59058, F3, 6) (dual of [59058, 59017, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(341, 59060, F3, 6) (dual of [59060, 59019, 7]-code), using
- 1 times truncation [i] based on linear OA(342, 59061, F3, 7) (dual of [59061, 59019, 8]-code), using
- construction X4 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(311, 12, F3, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,3)), using
- dual of repetition code with length 12 [i]
- linear OA(31, 12, F3, 1) (dual of [12, 11, 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(4) [i] based on
- 1 times truncation [i] based on linear OA(342, 59061, F3, 7) (dual of [59061, 59019, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(341, 59060, F3, 6) (dual of [59060, 59019, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(341, 59058, F3, 6) (dual of [59058, 59017, 7]-code), using
- appending kth column [i] based on linear OOA(341, 19686, F3, 5, 6) (dual of [(19686, 5), 98389, 7]-NRT-code), using
(41−6, 41, 59060)-Net over F3 — Digital
Digital (35, 41, 59060)-net over F3, using
- net defined by OOA [i] based on linear OOA(341, 59060, F3, 6, 6) (dual of [(59060, 6), 354319, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(341, 59060, F3, 5, 6) (dual of [(59060, 5), 295259, 7]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(341, 59060, F3, 6) (dual of [59060, 59019, 7]-code), using
- 1 times truncation [i] based on linear OA(342, 59061, F3, 7) (dual of [59061, 59019, 8]-code), using
- construction X4 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(311, 12, F3, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,3)), using
- dual of repetition code with length 12 [i]
- linear OA(31, 12, F3, 1) (dual of [12, 11, 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(4) [i] based on
- 1 times truncation [i] based on linear OA(342, 59061, F3, 7) (dual of [59061, 59019, 8]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(341, 59060, F3, 6) (dual of [59060, 59019, 7]-code), using
- appending kth column [i] based on linear OOA(341, 59060, F3, 5, 6) (dual of [(59060, 5), 295259, 7]-NRT-code), using
(41−6, 41, 3013079)-Net in Base 3 — Upper bound on s
There is no (35, 41, 3013080)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 36 473026 807101 871841 > 341 [i]