Best Known (37, 37+7, s)-Nets in Base 4
(37, 37+7, 21852)-Net over F4 — Constructive and digital
Digital (37, 44, 21852)-net over F4, using
- net defined by OOA [i] based on linear OOA(444, 21852, F4, 9, 7) (dual of [(21852, 9), 196624, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(444, 21853, F4, 3, 7) (dual of [(21853, 3), 65515, 8]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(43, 5, F4, 3, 3) (dual of [(5, 3), 12, 4]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(3;12,4) [i]
- linear OOA(441, 21848, F4, 3, 7) (dual of [(21848, 3), 65503, 8]-NRT-code), using
- OOA 3-folding [i] based on linear OA(441, 65544, F4, 7) (dual of [65544, 65503, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(441, 65536, F4, 7) (dual of [65536, 65495, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(433, 65536, F4, 6) (dual of [65536, 65503, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(6) ⊂ Ce(5) [i] based on
- OOA 3-folding [i] based on linear OA(441, 65544, F4, 7) (dual of [65544, 65503, 8]-code), using
- linear OOA(43, 5, F4, 3, 3) (dual of [(5, 3), 12, 4]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(444, 21853, F4, 3, 7) (dual of [(21853, 3), 65515, 8]-NRT-code), using
(37, 37+7, 65558)-Net over F4 — Digital
Digital (37, 44, 65558)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(444, 65558, F4, 7) (dual of [65558, 65514, 8]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(442, 65554, F4, 7) (dual of [65554, 65512, 8]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(441, 65536, F4, 7) (dual of [65536, 65495, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(425, 65536, F4, 5) (dual of [65536, 65511, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(417, 18, F4, 17) (dual of [18, 1, 18]-code or 18-arc in PG(16,4)), using
- dual of repetition code with length 18 [i]
- linear OA(41, 18, F4, 1) (dual of [18, 17, 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(6) ⊂ Ce(4) [i] based on
- linear OA(442, 65556, F4, 6) (dual of [65556, 65514, 7]-code), using Gilbert–Varšamov bound and bm = 442 > Vbs−1(k−1) = 2 451313 900934 462172 681304 [i]
- linear OA(40, 2, F4, 0) (dual of [2, 2, 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
- linear OA(442, 65554, F4, 7) (dual of [65554, 65512, 8]-code), using
- construction X with Varšamov bound [i] based on
(37, 37+7, large)-Net in Base 4 — Upper bound on s
There is no (37, 44, large)-net in base 4, because
- 5 times m-reduction [i] would yield (37, 39, large)-net in base 4, but