Best Known (33−6, 33, s)-Nets in Base 3
(33−6, 33, 2190)-Net over F3 — Constructive and digital
Digital (27, 33, 2190)-net over F3, using
- net defined by OOA [i] based on linear OOA(333, 2190, F3, 6, 6) (dual of [(2190, 6), 13107, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(333, 2190, F3, 5, 6) (dual of [(2190, 5), 10917, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(333, 6570, F3, 6) (dual of [6570, 6537, 7]-code), using
- 1 times truncation [i] based on linear OA(334, 6571, F3, 7) (dual of [6571, 6537, 8]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(333, 6561, F3, 7) (dual of [6561, 6528, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(325, 6561, F3, 5) (dual of [6561, 6536, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(39, 10, F3, 9) (dual of [10, 1, 10]-code or 10-arc in PG(8,3)), using
- dual of repetition code with length 10 [i]
- linear OA(31, 10, F3, 1) (dual of [10, 9, 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(334, 6571, F3, 7) (dual of [6571, 6537, 8]-code), using
- OA 3-folding and stacking [i] based on linear OA(333, 6570, F3, 6) (dual of [6570, 6537, 7]-code), using
- appending kth column [i] based on linear OOA(333, 2190, F3, 5, 6) (dual of [(2190, 5), 10917, 7]-NRT-code), using
(33−6, 33, 6570)-Net over F3 — Digital
Digital (27, 33, 6570)-net over F3, using
- net defined by OOA [i] based on linear OOA(333, 6570, F3, 6, 6) (dual of [(6570, 6), 39387, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(333, 6570, F3, 5, 6) (dual of [(6570, 5), 32817, 7]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(333, 6570, F3, 6) (dual of [6570, 6537, 7]-code), using
- 1 times truncation [i] based on linear OA(334, 6571, F3, 7) (dual of [6571, 6537, 8]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(333, 6561, F3, 7) (dual of [6561, 6528, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(325, 6561, F3, 5) (dual of [6561, 6536, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(39, 10, F3, 9) (dual of [10, 1, 10]-code or 10-arc in PG(8,3)), using
- dual of repetition code with length 10 [i]
- linear OA(31, 10, F3, 1) (dual of [10, 9, 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(334, 6571, F3, 7) (dual of [6571, 6537, 8]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(333, 6570, F3, 6) (dual of [6570, 6537, 7]-code), using
- appending kth column [i] based on linear OOA(333, 6570, F3, 5, 6) (dual of [(6570, 5), 32817, 7]-NRT-code), using
(33−6, 33, 160946)-Net in Base 3 — Upper bound on s
There is no (27, 33, 160947)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 5559 140265 192811 > 333 [i]