Best Known (78, 78+13, s)-Nets in Base 4
(78, 78+13, 174762)-Net over F4 — Constructive and digital
Digital (78, 91, 174762)-net over F4, using
- net defined by OOA [i] based on linear OOA(491, 174762, F4, 13, 13) (dual of [(174762, 13), 2271815, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(491, 1048573, F4, 13) (dual of [1048573, 1048482, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(491, 1048576, F4, 13) (dual of [1048576, 1048485, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(491, 1048576, F4, 13) (dual of [1048576, 1048485, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(491, 1048573, F4, 13) (dual of [1048573, 1048482, 14]-code), using
(78, 78+13, 349525)-Net over F4 — Digital
Digital (78, 91, 349525)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(491, 349525, F4, 3, 13) (dual of [(349525, 3), 1048484, 14]-NRT-code), using
- OOA 3-folding [i] based on linear OA(491, 1048575, F4, 13) (dual of [1048575, 1048484, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(491, 1048576, F4, 13) (dual of [1048576, 1048485, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(491, 1048576, F4, 13) (dual of [1048576, 1048485, 14]-code), using
- OOA 3-folding [i] based on linear OA(491, 1048575, F4, 13) (dual of [1048575, 1048484, 14]-code), using
(78, 78+13, large)-Net in Base 4 — Upper bound on s
There is no (78, 91, large)-net in base 4, because
- 11 times m-reduction [i] would yield (78, 80, large)-net in base 4, but