Best Known (72, 86, s)-Nets in Base 4
(72, 86, 9366)-Net over F4 — Constructive and digital
Digital (72, 86, 9366)-net over F4, using
- net defined by OOA [i] based on linear OOA(486, 9366, F4, 14, 14) (dual of [(9366, 14), 131038, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(486, 65562, F4, 14) (dual of [65562, 65476, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(486, 65565, F4, 14) (dual of [65565, 65479, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- 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(457, 65536, F4, 10) (dual of [65536, 65479, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(13) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(486, 65565, F4, 14) (dual of [65565, 65479, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(486, 65562, F4, 14) (dual of [65562, 65476, 15]-code), using
(72, 86, 32782)-Net over F4 — Digital
Digital (72, 86, 32782)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(486, 32782, F4, 2, 14) (dual of [(32782, 2), 65478, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(486, 65564, F4, 14) (dual of [65564, 65478, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(486, 65565, F4, 14) (dual of [65565, 65479, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- 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(457, 65536, F4, 10) (dual of [65536, 65479, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(13) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(486, 65565, F4, 14) (dual of [65565, 65479, 15]-code), using
- OOA 2-folding [i] based on linear OA(486, 65564, F4, 14) (dual of [65564, 65478, 15]-code), using
(72, 86, large)-Net in Base 4 — Upper bound on s
There is no (72, 86, large)-net in base 4, because
- 12 times m-reduction [i] would yield (72, 74, large)-net in base 4, but