Best Known (76−20, 76, s)-Nets in Base 16
(76−20, 76, 6555)-Net over F16 — Constructive and digital
Digital (56, 76, 6555)-net over F16, using
- 161 times duplication [i] based on digital (55, 75, 6555)-net over F16, using
- net defined by OOA [i] based on linear OOA(1675, 6555, F16, 20, 20) (dual of [(6555, 20), 131025, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(1675, 65550, F16, 20) (dual of [65550, 65475, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(16) [i] based on
- linear OA(1673, 65536, F16, 20) (dual of [65536, 65463, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(162, 14, F16, 2) (dual of [14, 12, 3]-code or 14-arc in PG(1,16)), using
- discarding factors / shortening the dual code based on linear OA(162, 16, F16, 2) (dual of [16, 14, 3]-code or 16-arc in PG(1,16)), using
- Reed–Solomon code RS(14,16) [i]
- discarding factors / shortening the dual code based on linear OA(162, 16, F16, 2) (dual of [16, 14, 3]-code or 16-arc in PG(1,16)), using
- construction X applied to Ce(19) ⊂ Ce(16) [i] based on
- OA 10-folding and stacking [i] based on linear OA(1675, 65550, F16, 20) (dual of [65550, 65475, 21]-code), using
- net defined by OOA [i] based on linear OOA(1675, 6555, F16, 20, 20) (dual of [(6555, 20), 131025, 21]-NRT-code), using
(76−20, 76, 52376)-Net over F16 — Digital
Digital (56, 76, 52376)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1676, 52376, F16, 20) (dual of [52376, 52300, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(1676, 65551, F16, 20) (dual of [65551, 65475, 21]-code), using
- 1 times code embedding in larger space [i] based on linear OA(1675, 65550, F16, 20) (dual of [65550, 65475, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(16) [i] based on
- linear OA(1673, 65536, F16, 20) (dual of [65536, 65463, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(162, 14, F16, 2) (dual of [14, 12, 3]-code or 14-arc in PG(1,16)), using
- discarding factors / shortening the dual code based on linear OA(162, 16, F16, 2) (dual of [16, 14, 3]-code or 16-arc in PG(1,16)), using
- Reed–Solomon code RS(14,16) [i]
- discarding factors / shortening the dual code based on linear OA(162, 16, F16, 2) (dual of [16, 14, 3]-code or 16-arc in PG(1,16)), using
- construction X applied to Ce(19) ⊂ Ce(16) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(1675, 65550, F16, 20) (dual of [65550, 65475, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(1676, 65551, F16, 20) (dual of [65551, 65475, 21]-code), using
(76−20, 76, large)-Net in Base 16 — Upper bound on s
There is no (56, 76, large)-net in base 16, because
- 18 times m-reduction [i] would yield (56, 58, large)-net in base 16, but