Best Known (87, 87+16, s)-Nets in Base 4
(87, 87+16, 8195)-Net over F4 — Constructive and digital
Digital (87, 103, 8195)-net over F4, using
- 41 times duplication [i] based on digital (86, 102, 8195)-net over F4, using
- t-expansion [i] based on digital (85, 102, 8195)-net over F4, using
- net defined by OOA [i] based on linear OOA(4102, 8195, F4, 17, 17) (dual of [(8195, 17), 139213, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(4102, 65561, F4, 17) (dual of [65561, 65459, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(4102, 65565, F4, 17) (dual of [65565, 65463, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- linear OA(497, 65536, F4, 17) (dual of [65536, 65439, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(473, 65536, F4, 13) (dual of [65536, 65463, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(45, 29, F4, 3) (dual of [29, 24, 4]-code or 29-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(16) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(4102, 65565, F4, 17) (dual of [65565, 65463, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(4102, 65561, F4, 17) (dual of [65561, 65459, 18]-code), using
- net defined by OOA [i] based on linear OOA(4102, 8195, F4, 17, 17) (dual of [(8195, 17), 139213, 18]-NRT-code), using
- t-expansion [i] based on digital (85, 102, 8195)-net over F4, using
(87, 87+16, 49055)-Net over F4 — Digital
Digital (87, 103, 49055)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4103, 49055, F4, 16) (dual of [49055, 48952, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(4103, 65567, F4, 16) (dual of [65567, 65464, 17]-code), using
- 1 times truncation [i] based on linear OA(4104, 65568, F4, 17) (dual of [65568, 65464, 18]-code), using
- construction XX applied to Ce(16) ⊂ Ce(12) ⊂ Ce(10) [i] based on
- linear OA(497, 65536, F4, 17) (dual of [65536, 65439, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(473, 65536, F4, 13) (dual of [65536, 65463, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(465, 65536, F4, 11) (dual of [65536, 65471, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(45, 30, F4, 3) (dual of [30, 25, 4]-code or 30-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
- linear OA(41, 2, F4, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(16) ⊂ Ce(12) ⊂ Ce(10) [i] based on
- 1 times truncation [i] based on linear OA(4104, 65568, F4, 17) (dual of [65568, 65464, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(4103, 65567, F4, 16) (dual of [65567, 65464, 17]-code), using
(87, 87+16, large)-Net in Base 4 — Upper bound on s
There is no (87, 103, large)-net in base 4, because
- 14 times m-reduction [i] would yield (87, 89, large)-net in base 4, but