Best Known (82−22, 82, s)-Nets in Base 16
(82−22, 82, 5958)-Net over F16 — Constructive and digital
Digital (60, 82, 5958)-net over F16, using
- 161 times duplication [i] based on digital (59, 81, 5958)-net over F16, using
- net defined by OOA [i] based on linear OOA(1681, 5958, F16, 22, 22) (dual of [(5958, 22), 130995, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(1681, 65538, F16, 22) (dual of [65538, 65457, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(1681, 65540, F16, 22) (dual of [65540, 65459, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- linear OA(1681, 65536, F16, 22) (dual of [65536, 65455, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- 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(160, 4, F16, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(1681, 65540, F16, 22) (dual of [65540, 65459, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(1681, 65538, F16, 22) (dual of [65538, 65457, 23]-code), using
- net defined by OOA [i] based on linear OOA(1681, 5958, F16, 22, 22) (dual of [(5958, 22), 130995, 23]-NRT-code), using
(82−22, 82, 41667)-Net over F16 — Digital
Digital (60, 82, 41667)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1682, 41667, F16, 22) (dual of [41667, 41585, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(1682, 65545, F16, 22) (dual of [65545, 65463, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(19) [i] based on
- linear OA(1681, 65536, F16, 22) (dual of [65536, 65455, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- 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(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(21) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(1682, 65545, F16, 22) (dual of [65545, 65463, 23]-code), using
(82−22, 82, large)-Net in Base 16 — Upper bound on s
There is no (60, 82, large)-net in base 16, because
- 20 times m-reduction [i] would yield (60, 62, large)-net in base 16, but