Best Known (86, 86+18, s)-Nets in Base 4
(86, 86+18, 1835)-Net over F4 — Constructive and digital
Digital (86, 104, 1835)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (3, 12, 14)-net over F4, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 3 and N(F) ≥ 14, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- digital (74, 92, 1821)-net over F4, using
- net defined by OOA [i] based on linear OOA(492, 1821, F4, 18, 18) (dual of [(1821, 18), 32686, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(492, 16389, F4, 18) (dual of [16389, 16297, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(492, 16391, F4, 18) (dual of [16391, 16299, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(492, 16384, F4, 18) (dual of [16384, 16292, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(485, 16384, F4, 17) (dual of [16384, 16299, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(40, 7, F4, 0) (dual of [7, 7, 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(17) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(492, 16391, F4, 18) (dual of [16391, 16299, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(492, 16389, F4, 18) (dual of [16389, 16297, 19]-code), using
- net defined by OOA [i] based on linear OOA(492, 1821, F4, 18, 18) (dual of [(1821, 18), 32686, 19]-NRT-code), using
- digital (3, 12, 14)-net over F4, using
(86, 86+18, 16430)-Net over F4 — Digital
Digital (86, 104, 16430)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4104, 16430, F4, 18) (dual of [16430, 16326, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(10) [i] based on
- linear OA(492, 16384, F4, 18) (dual of [16384, 16292, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(457, 16384, F4, 11) (dual of [16384, 16327, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(412, 46, F4, 6) (dual of [46, 34, 7]-code), using
- a “GraQC†code from Grassl’s database [i]
- construction X applied to Ce(17) ⊂ Ce(10) [i] based on
(86, 86+18, large)-Net in Base 4 — Upper bound on s
There is no (86, 104, large)-net in base 4, because
- 16 times m-reduction [i] would yield (86, 88, large)-net in base 4, but