Best Known (78−10, 78, s)-Nets in Base 4
(78−10, 78, 838863)-Net over F4 — Constructive and digital
Digital (68, 78, 838863)-net over F4, using
- net defined by OOA [i] based on linear OOA(478, 838863, F4, 10, 10) (dual of [(838863, 10), 8388552, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(478, 4194315, F4, 10) (dual of [4194315, 4194237, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(478, 4194304, F4, 10) (dual of [4194304, 4194226, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(467, 4194304, F4, 9) (dual of [4194304, 4194237, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(40, 11, F4, 0) (dual of [11, 11, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- OA 5-folding and stacking [i] based on linear OA(478, 4194315, F4, 10) (dual of [4194315, 4194237, 11]-code), using
(78−10, 78, 2097157)-Net over F4 — Digital
Digital (68, 78, 2097157)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(478, 2097157, F4, 2, 10) (dual of [(2097157, 2), 4194236, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(478, 4194314, F4, 10) (dual of [4194314, 4194236, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(478, 4194315, F4, 10) (dual of [4194315, 4194237, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(478, 4194304, F4, 10) (dual of [4194304, 4194226, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(467, 4194304, F4, 9) (dual of [4194304, 4194237, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(40, 11, F4, 0) (dual of [11, 11, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(478, 4194315, F4, 10) (dual of [4194315, 4194237, 11]-code), using
- OOA 2-folding [i] based on linear OA(478, 4194314, F4, 10) (dual of [4194314, 4194236, 11]-code), using
(78−10, 78, large)-Net in Base 4 — Upper bound on s
There is no (68, 78, large)-net in base 4, because
- 8 times m-reduction [i] would yield (68, 70, large)-net in base 4, but