Best Known (81−21, 81, s)-Nets in Base 16
(81−21, 81, 6555)-Net over F16 — Constructive and digital
Digital (60, 81, 6555)-net over F16, using
- 161 times duplication [i] based on digital (59, 80, 6555)-net over F16, using
- net defined by OOA [i] based on linear OOA(1680, 6555, F16, 21, 21) (dual of [(6555, 21), 137575, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(1680, 65551, F16, 21) (dual of [65551, 65471, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(1680, 65554, F16, 21) (dual of [65554, 65474, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(16) [i] based on
- linear OA(1677, 65536, F16, 21) (dual of [65536, 65459, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(1661, 65536, F16, 17) (dual of [65536, 65475, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(163, 18, F16, 3) (dual of [18, 15, 4]-code or 18-arc in PG(2,16) or 18-cap in PG(2,16)), using
- construction X applied to Ce(20) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(1680, 65554, F16, 21) (dual of [65554, 65474, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(1680, 65551, F16, 21) (dual of [65551, 65471, 22]-code), using
- net defined by OOA [i] based on linear OOA(1680, 6555, F16, 21, 21) (dual of [(6555, 21), 137575, 22]-NRT-code), using
(81−21, 81, 62092)-Net over F16 — Digital
Digital (60, 81, 62092)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1681, 62092, F16, 21) (dual of [62092, 62011, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(1681, 65556, F16, 21) (dual of [65556, 65475, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(16) [i] based on
- linear OA(1677, 65536, F16, 21) (dual of [65536, 65459, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(1661, 65536, F16, 17) (dual of [65536, 65475, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(164, 20, F16, 3) (dual of [20, 16, 4]-code or 20-cap in PG(3,16)), using
- construction X applied to Ce(20) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(1681, 65556, F16, 21) (dual of [65556, 65475, 22]-code), using
(81−21, 81, large)-Net in Base 16 — Upper bound on s
There is no (60, 81, large)-net in base 16, because
- 19 times m-reduction [i] would yield (60, 62, large)-net in base 16, but