Best Known (59, 59+10, s)-Nets in Base 4
(59, 59+10, 52435)-Net over F4 — Constructive and digital
Digital (59, 69, 52435)-net over F4, using
- net defined by OOA [i] based on linear OOA(469, 52435, F4, 10, 10) (dual of [(52435, 10), 524281, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(469, 262175, F4, 10) (dual of [262175, 262106, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(469, 262176, F4, 10) (dual of [262176, 262107, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(5) [i] based on
- linear OA(464, 262144, F4, 10) (dual of [262144, 262080, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(437, 262144, F4, 6) (dual of [262144, 262107, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(9) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(469, 262176, F4, 10) (dual of [262176, 262107, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(469, 262175, F4, 10) (dual of [262175, 262106, 11]-code), using
(59, 59+10, 164461)-Net over F4 — Digital
Digital (59, 69, 164461)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(469, 164461, F4, 10) (dual of [164461, 164392, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(469, 262150, F4, 10) (dual of [262150, 262081, 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(464, 262144, F4, 10) (dual of [262144, 262080, 11]-code), using
- an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, 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(469, 262150, F4, 10) (dual of [262150, 262081, 11]-code), using
(59, 59+10, large)-Net in Base 4 — Upper bound on s
There is no (59, 69, large)-net in base 4, because
- 8 times m-reduction [i] would yield (59, 61, large)-net in base 4, but