Best Known (67, 78, s)-Nets in Base 4
(67, 78, 52435)-Net over F4 — Constructive and digital
Digital (67, 78, 52435)-net over F4, using
- net defined by OOA [i] based on linear OOA(478, 52435, F4, 11, 11) (dual of [(52435, 11), 576707, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(478, 262176, F4, 11) (dual of [262176, 262098, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(6) [i] based on
- linear OA(473, 262144, F4, 11) (dual of [262144, 262071, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(446, 262144, F4, 7) (dual of [262144, 262098, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(45, 32, F4, 3) (dual of [32, 27, 4]-code or 32-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(478, 262176, F4, 11) (dual of [262176, 262098, 12]-code), using
(67, 78, 195692)-Net over F4 — Digital
Digital (67, 78, 195692)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(478, 195692, F4, 11) (dual of [195692, 195614, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(478, 262176, F4, 11) (dual of [262176, 262098, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(6) [i] based on
- linear OA(473, 262144, F4, 11) (dual of [262144, 262071, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(446, 262144, F4, 7) (dual of [262144, 262098, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(45, 32, F4, 3) (dual of [32, 27, 4]-code or 32-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(478, 262176, F4, 11) (dual of [262176, 262098, 12]-code), using
(67, 78, large)-Net in Base 4 — Upper bound on s
There is no (67, 78, large)-net in base 4, because
- 9 times m-reduction [i] would yield (67, 69, large)-net in base 4, but