Best Known (96−12, 96, s)-Nets in Base 4
(96−12, 96, 174768)-Net over F4 — Constructive and digital
Digital (84, 96, 174768)-net over F4, using
- t-expansion [i] based on digital (83, 96, 174768)-net over F4, using
- net defined by OOA [i] based on linear OOA(496, 174768, F4, 13, 13) (dual of [(174768, 13), 2271888, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(496, 1048609, F4, 13) (dual of [1048609, 1048513, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(496, 1048611, F4, 13) (dual of [1048611, 1048515, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] 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]
- linear OA(461, 1048576, F4, 9) (dual of [1048576, 1048515, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(45, 35, F4, 3) (dual of [35, 30, 4]-code or 35-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(496, 1048611, F4, 13) (dual of [1048611, 1048515, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(496, 1048609, F4, 13) (dual of [1048609, 1048513, 14]-code), using
- net defined by OOA [i] based on linear OOA(496, 174768, F4, 13, 13) (dual of [(174768, 13), 2271888, 14]-NRT-code), using
(96−12, 96, 791445)-Net over F4 — Digital
Digital (84, 96, 791445)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(496, 791445, F4, 12) (dual of [791445, 791349, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(496, 1048611, F4, 12) (dual of [1048611, 1048515, 13]-code), using
- strength reduction [i] based on linear OA(496, 1048611, F4, 13) (dual of [1048611, 1048515, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] 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]
- linear OA(461, 1048576, F4, 9) (dual of [1048576, 1048515, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(45, 35, F4, 3) (dual of [35, 30, 4]-code or 35-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- strength reduction [i] based on linear OA(496, 1048611, F4, 13) (dual of [1048611, 1048515, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(496, 1048611, F4, 12) (dual of [1048611, 1048515, 13]-code), using
(96−12, 96, large)-Net in Base 4 — Upper bound on s
There is no (84, 96, large)-net in base 4, because
- 10 times m-reduction [i] would yield (84, 86, large)-net in base 4, but