Best Known (40−5, 40, s)-Nets in Base 3
(40−5, 40, 797167)-Net over F3 — Constructive and digital
Digital (35, 40, 797167)-net over F3, using
- net defined by OOA [i] based on linear OOA(340, 797167, F3, 5, 5) (dual of [(797167, 5), 3985795, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(340, 1594335, F3, 5) (dual of [1594335, 1594295, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(340, 1594336, F3, 5) (dual of [1594336, 1594296, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(340, 1594323, F3, 5) (dual of [1594323, 1594283, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(327, 1594323, F3, 4) (dual of [1594323, 1594296, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(30, 13, F3, 0) (dual of [13, 13, 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
- discarding factors / shortening the dual code based on linear OA(340, 1594336, F3, 5) (dual of [1594336, 1594296, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(340, 1594335, F3, 5) (dual of [1594335, 1594295, 6]-code), using
(40−5, 40, 1448537)-Net over F3 — Digital
Digital (35, 40, 1448537)-net over F3, using
- net defined by OOA [i] based on linear OOA(340, 1448537, F3, 5, 5) (dual of [(1448537, 5), 7242645, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(340, 1448537, F3, 4, 5) (dual of [(1448537, 4), 5794108, 6]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(340, 1448537, F3, 5) (dual of [1448537, 1448497, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(340, 1594323, F3, 5) (dual of [1594323, 1594283, 6]-code), using
- an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(340, 1594323, F3, 5) (dual of [1594323, 1594283, 6]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(340, 1448537, F3, 5) (dual of [1448537, 1448497, 6]-code), using
- appending kth column [i] based on linear OOA(340, 1448537, F3, 4, 5) (dual of [(1448537, 4), 5794108, 6]-NRT-code), using
(40−5, 40, large)-Net in Base 3 — Upper bound on s
There is no (35, 40, large)-net in base 3, because
- 3 times m-reduction [i] would yield (35, 37, large)-net in base 3, but