Best Known (66−10, 66, s)-Nets in Base 5
(66−10, 66, 78128)-Net over F5 — Constructive and digital
Digital (56, 66, 78128)-net over F5, using
- net defined by OOA [i] based on linear OOA(566, 78128, F5, 10, 10) (dual of [(78128, 10), 781214, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(566, 390640, F5, 10) (dual of [390640, 390574, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(566, 390642, F5, 10) (dual of [390642, 390576, 11]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(565, 390625, F5, 11) (dual of [390625, 390560, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(51, 17, F5, 1) (dual of [17, 16, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(566, 390642, F5, 10) (dual of [390642, 390576, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(566, 390640, F5, 10) (dual of [390640, 390574, 11]-code), using
(66−10, 66, 390643)-Net over F5 — Digital
Digital (56, 66, 390643)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(566, 390643, F5, 10) (dual of [390643, 390577, 11]-code), using
- construction X4 applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(565, 390625, F5, 11) (dual of [390625, 390560, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(517, 18, F5, 17) (dual of [18, 1, 18]-code or 18-arc in PG(16,5)), using
- dual of repetition code with length 18 [i]
- linear OA(51, 18, F5, 1) (dual of [18, 17, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(10) ⊂ Ce(7) [i] based on
(66−10, 66, large)-Net in Base 5 — Upper bound on s
There is no (56, 66, large)-net in base 5, because
- 8 times m-reduction [i] would yield (56, 58, large)-net in base 5, but