Best Known (87, 87+15, s)-Nets in Base 4
(87, 87+15, 37452)-Net over F4 — Constructive and digital
Digital (87, 102, 37452)-net over F4, using
- net defined by OOA [i] based on linear OOA(4102, 37452, F4, 15, 15) (dual of [(37452, 15), 561678, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(4102, 262165, F4, 15) (dual of [262165, 262063, 16]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4101, 262164, F4, 15) (dual of [262164, 262063, 16]-code), using
- construction X4 applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(4100, 262144, F4, 15) (dual of [262144, 262044, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(482, 262144, F4, 13) (dual of [262144, 262062, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(419, 20, F4, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,4)), using
- dual of repetition code with length 20 [i]
- linear OA(41, 20, F4, 1) (dual of [20, 19, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(14) ⊂ Ce(12) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4101, 262164, F4, 15) (dual of [262164, 262063, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(4102, 262165, F4, 15) (dual of [262165, 262063, 16]-code), using
(87, 87+15, 131082)-Net over F4 — Digital
Digital (87, 102, 131082)-net over F4, using
- 41 times duplication [i] based on digital (86, 101, 131082)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4101, 131082, F4, 2, 15) (dual of [(131082, 2), 262063, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4101, 262164, F4, 15) (dual of [262164, 262063, 16]-code), using
- construction X4 applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(4100, 262144, F4, 15) (dual of [262144, 262044, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(482, 262144, F4, 13) (dual of [262144, 262062, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(419, 20, F4, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,4)), using
- dual of repetition code with length 20 [i]
- linear OA(41, 20, F4, 1) (dual of [20, 19, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(14) ⊂ Ce(12) [i] based on
- OOA 2-folding [i] based on linear OA(4101, 262164, F4, 15) (dual of [262164, 262063, 16]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4101, 131082, F4, 2, 15) (dual of [(131082, 2), 262063, 16]-NRT-code), using
(87, 87+15, large)-Net in Base 4 — Upper bound on s
There is no (87, 102, large)-net in base 4, because
- 13 times m-reduction [i] would yield (87, 89, large)-net in base 4, but