Best Known (76−10, 76, s)-Nets in Base 4
(76−10, 76, 209722)-Net over F4 — Constructive and digital
Digital (66, 76, 209722)-net over F4, using
- net defined by OOA [i] based on linear OOA(476, 209722, F4, 10, 10) (dual of [(209722, 10), 2097144, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(476, 1048610, F4, 10) (dual of [1048610, 1048534, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(476, 1048611, F4, 10) (dual of [1048611, 1048535, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(5) [i] based on
- linear OA(471, 1048576, F4, 10) (dual of [1048576, 1048505, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(441, 1048576, F4, 6) (dual of [1048576, 1048535, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(9) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(476, 1048611, F4, 10) (dual of [1048611, 1048535, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(476, 1048610, F4, 10) (dual of [1048610, 1048534, 11]-code), using
(76−10, 76, 553193)-Net over F4 — Digital
Digital (66, 76, 553193)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(476, 553193, F4, 10) (dual of [553193, 553117, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(476, 1048582, F4, 10) (dual of [1048582, 1048506, 11]-code), using
- (u, u+v)-construction [i] based on
- linear OA(45, 6, F4, 5) (dual of [6, 1, 6]-code or 6-arc in PG(4,4)), using
- dual of repetition code with length 6 [i]
- linear OA(471, 1048576, F4, 10) (dual of [1048576, 1048505, 11]-code), using
- an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(45, 6, F4, 5) (dual of [6, 1, 6]-code or 6-arc in PG(4,4)), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(476, 1048582, F4, 10) (dual of [1048582, 1048506, 11]-code), using
(76−10, 76, large)-Net in Base 4 — Upper bound on s
There is no (66, 76, large)-net in base 4, because
- 8 times m-reduction [i] would yield (66, 68, large)-net in base 4, but