Best Known (78, 78+14, s)-Nets in Base 5
(78, 78+14, 55807)-Net over F5 — Constructive and digital
Digital (78, 92, 55807)-net over F5, using
- net defined by OOA [i] based on linear OOA(592, 55807, F5, 14, 14) (dual of [(55807, 14), 781206, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(592, 390649, F5, 14) (dual of [390649, 390557, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(592, 390652, F5, 14) (dual of [390652, 390560, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- linear OA(589, 390625, F5, 14) (dual of [390625, 390536, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- 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(53, 27, F5, 2) (dual of [27, 24, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(53, 31, F5, 2) (dual of [31, 28, 3]-code), using
- Hamming code H(3,5) [i]
- discarding factors / shortening the dual code based on linear OA(53, 31, F5, 2) (dual of [31, 28, 3]-code), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(592, 390652, F5, 14) (dual of [390652, 390560, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(592, 390649, F5, 14) (dual of [390649, 390557, 15]-code), using
(78, 78+14, 264126)-Net over F5 — Digital
Digital (78, 92, 264126)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(592, 264126, F5, 14) (dual of [264126, 264034, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(592, 390652, F5, 14) (dual of [390652, 390560, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- linear OA(589, 390625, F5, 14) (dual of [390625, 390536, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- 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(53, 27, F5, 2) (dual of [27, 24, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(53, 31, F5, 2) (dual of [31, 28, 3]-code), using
- Hamming code H(3,5) [i]
- discarding factors / shortening the dual code based on linear OA(53, 31, F5, 2) (dual of [31, 28, 3]-code), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(592, 390652, F5, 14) (dual of [390652, 390560, 15]-code), using
(78, 78+14, large)-Net in Base 5 — Upper bound on s
There is no (78, 92, large)-net in base 5, because
- 12 times m-reduction [i] would yield (78, 80, large)-net in base 5, but