Best Known (87, 87+31, s)-Nets in Base 16
(87, 87+31, 4369)-Net over F16 — Constructive and digital
Digital (87, 118, 4369)-net over F16, using
- 161 times duplication [i] based on digital (86, 117, 4369)-net over F16, using
- net defined by OOA [i] based on linear OOA(16117, 4369, F16, 31, 31) (dual of [(4369, 31), 135322, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(16117, 65536, F16, 31) (dual of [65536, 65419, 32]-code), using
- an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- OOA 15-folding and stacking with additional row [i] based on linear OA(16117, 65536, F16, 31) (dual of [65536, 65419, 32]-code), using
- net defined by OOA [i] based on linear OOA(16117, 4369, F16, 31, 31) (dual of [(4369, 31), 135322, 32]-NRT-code), using
(87, 87+31, 56094)-Net over F16 — Digital
Digital (87, 118, 56094)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16118, 56094, F16, 31) (dual of [56094, 55976, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(16118, 65545, F16, 31) (dual of [65545, 65427, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(16117, 65536, F16, 31) (dual of [65536, 65419, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(16109, 65536, F16, 29) (dual of [65536, 65427, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(161, 9, F16, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(161, s, F16, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(16118, 65545, F16, 31) (dual of [65545, 65427, 32]-code), using
(87, 87+31, large)-Net in Base 16 — Upper bound on s
There is no (87, 118, large)-net in base 16, because
- 29 times m-reduction [i] would yield (87, 89, large)-net in base 16, but