Best Known (77−20, 77, s)-Nets in Base 16
(77−20, 77, 6555)-Net over F16 — Constructive and digital
Digital (57, 77, 6555)-net over F16, using
- 162 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
(77−20, 77, 61099)-Net over F16 — Digital
Digital (57, 77, 61099)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1677, 61099, F16, 20) (dual of [61099, 61022, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(1677, 65553, F16, 20) (dual of [65553, 65476, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(14) [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(1657, 65536, F16, 15) (dual of [65536, 65479, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(164, 17, F16, 4) (dual of [17, 13, 5]-code or 17-arc in PG(3,16)), using
- extended Reed–Solomon code RSe(13,16) [i]
- construction X applied to Ce(19) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(1677, 65553, F16, 20) (dual of [65553, 65476, 21]-code), using
(77−20, 77, large)-Net in Base 16 — Upper bound on s
There is no (57, 77, large)-net in base 16, because
- 18 times m-reduction [i] would yield (57, 59, large)-net in base 16, but