Best Known (84, 84+15, s)-Nets in Base 4
(84, 84+15, 9378)-Net over F4 — Constructive and digital
Digital (84, 99, 9378)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (3, 10, 15)-net over F4, using
- digital (74, 89, 9363)-net over F4, using
- net defined by OOA [i] based on linear OOA(489, 9363, F4, 15, 15) (dual of [(9363, 15), 140356, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(489, 65542, F4, 15) (dual of [65542, 65453, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(489, 65544, F4, 15) (dual of [65544, 65455, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- linear OA(489, 65536, F4, 15) (dual of [65536, 65447, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(481, 65536, F4, 14) (dual of [65536, 65455, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(40, 8, F4, 0) (dual of [8, 8, 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(14) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(489, 65544, F4, 15) (dual of [65544, 65455, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(489, 65542, F4, 15) (dual of [65542, 65453, 16]-code), using
- net defined by OOA [i] based on linear OOA(489, 9363, F4, 15, 15) (dual of [(9363, 15), 140356, 16]-NRT-code), using
(84, 84+15, 65287)-Net over F4 — Digital
Digital (84, 99, 65287)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(499, 65287, F4, 15) (dual of [65287, 65188, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(499, 65556, F4, 15) (dual of [65556, 65457, 16]-code), using
- (u, u+v)-construction [i] based on
- linear OA(410, 20, F4, 7) (dual of [20, 10, 8]-code), using
- extended quadratic residue code Qe(20,4) [i]
- linear OA(489, 65536, F4, 15) (dual of [65536, 65447, 16]-code), using
- an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(410, 20, F4, 7) (dual of [20, 10, 8]-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(499, 65556, F4, 15) (dual of [65556, 65457, 16]-code), using
(84, 84+15, large)-Net in Base 4 — Upper bound on s
There is no (84, 99, large)-net in base 4, because
- 13 times m-reduction [i] would yield (84, 86, large)-net in base 4, but