Best Known (75, 86, s)-Nets in Base 4
(75, 86, 209722)-Net over F4 — Constructive and digital
Digital (75, 86, 209722)-net over F4, using
- net defined by OOA [i] based on linear OOA(486, 209722, F4, 11, 11) (dual of [(209722, 11), 2306856, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(486, 1048611, F4, 11) (dual of [1048611, 1048525, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(6) [i] based on
- linear OA(481, 1048576, F4, 11) (dual of [1048576, 1048495, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(451, 1048576, F4, 7) (dual of [1048576, 1048525, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(10) ⊂ Ce(6) [i] based on
- OOA 5-folding and stacking with additional row [i] based on linear OA(486, 1048611, F4, 11) (dual of [1048611, 1048525, 12]-code), using
(75, 86, 671040)-Net over F4 — Digital
Digital (75, 86, 671040)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(486, 671040, F4, 11) (dual of [671040, 670954, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(486, 1048611, F4, 11) (dual of [1048611, 1048525, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(6) [i] based on
- linear OA(481, 1048576, F4, 11) (dual of [1048576, 1048495, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(451, 1048576, F4, 7) (dual of [1048576, 1048525, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(10) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(486, 1048611, F4, 11) (dual of [1048611, 1048525, 12]-code), using
(75, 86, large)-Net in Base 4 — Upper bound on s
There is no (75, 86, large)-net in base 4, because
- 9 times m-reduction [i] would yield (75, 77, large)-net in base 4, but