Best Known (76, 89, s)-Nets in Base 5
(76, 89, 65119)-Net over F5 — Constructive and digital
Digital (76, 89, 65119)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (2, 8, 14)-net over F5, using
- digital (68, 81, 65105)-net over F5, using
- net defined by OOA [i] based on linear OOA(581, 65105, F5, 13, 13) (dual of [(65105, 13), 846284, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(581, 390631, F5, 13) (dual of [390631, 390550, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(581, 390633, F5, 13) (dual of [390633, 390552, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(581, 390625, F5, 13) (dual of [390625, 390544, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(573, 390625, F5, 12) (dual of [390625, 390552, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(50, 8, F5, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(581, 390633, F5, 13) (dual of [390633, 390552, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(581, 390631, F5, 13) (dual of [390631, 390550, 14]-code), using
- net defined by OOA [i] based on linear OOA(581, 65105, F5, 13, 13) (dual of [(65105, 13), 846284, 14]-NRT-code), using
(76, 89, 390666)-Net over F5 — Digital
Digital (76, 89, 390666)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(589, 390666, F5, 13) (dual of [390666, 390577, 14]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(588, 390664, F5, 13) (dual of [390664, 390576, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(7) [i] based on
- linear OA(581, 390625, F5, 13) (dual of [390625, 390544, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(549, 390625, F5, 8) (dual of [390625, 390576, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(57, 39, F5, 4) (dual of [39, 32, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(57, 44, F5, 4) (dual of [44, 37, 5]-code), using
- construction X applied to Ce(12) ⊂ Ce(7) [i] based on
- linear OA(588, 390665, F5, 12) (dual of [390665, 390577, 13]-code), using Gilbert–Varšamov bound and bm = 588 > Vbs−1(k−1) = 3 398469 823732 163958 679189 960723 656188 181951 483795 999341 157857 [i]
- linear OA(50, 1, F5, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(588, 390664, F5, 13) (dual of [390664, 390576, 14]-code), using
- construction X with Varšamov bound [i] based on
(76, 89, large)-Net in Base 5 — Upper bound on s
There is no (76, 89, large)-net in base 5, because
- 11 times m-reduction [i] would yield (76, 78, large)-net in base 5, but