Best Known (65, 81, s)-Nets in Base 16
(65, 81, 131075)-Net over F16 — Constructive and digital
Digital (65, 81, 131075)-net over F16, using
- 1 times m-reduction [i] based on digital (65, 82, 131075)-net over F16, using
- net defined by OOA [i] based on linear OOA(1682, 131075, F16, 17, 17) (dual of [(131075, 17), 2228193, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(1682, 1048601, F16, 17) (dual of [1048601, 1048519, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(1682, 1048602, F16, 17) (dual of [1048602, 1048520, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(11) [i] based on
- linear OA(1676, 1048576, F16, 17) (dual of [1048576, 1048500, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(1656, 1048576, F16, 12) (dual of [1048576, 1048520, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(166, 26, F16, 4) (dual of [26, 20, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(166, 240, F16, 4) (dual of [240, 234, 5]-code), using
- 1 times truncation [i] based on linear OA(167, 241, F16, 5) (dual of [241, 234, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(166, 240, F16, 4) (dual of [240, 234, 5]-code), using
- construction X applied to Ce(16) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(1682, 1048602, F16, 17) (dual of [1048602, 1048520, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(1682, 1048601, F16, 17) (dual of [1048601, 1048519, 18]-code), using
- net defined by OOA [i] based on linear OOA(1682, 131075, F16, 17, 17) (dual of [(131075, 17), 2228193, 18]-NRT-code), using
(65, 81, 262144)-Net in Base 16 — Constructive
(65, 81, 262144)-net in base 16, using
- net defined by OOA [i] based on OOA(1681, 262144, S16, 16, 16), using
- OA 8-folding and stacking [i] based on OA(1681, 2097152, S16, 16), using
- discarding factors based on OA(1681, 2097155, S16, 16), using
- discarding parts of the base [i] based on linear OA(12846, 2097155, F128, 16) (dual of [2097155, 2097109, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(14) [i] based on
- linear OA(12846, 2097152, F128, 16) (dual of [2097152, 2097106, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(12843, 2097152, F128, 15) (dual of [2097152, 2097109, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(15) ⊂ Ce(14) [i] based on
- discarding parts of the base [i] based on linear OA(12846, 2097155, F128, 16) (dual of [2097155, 2097109, 17]-code), using
- discarding factors based on OA(1681, 2097155, S16, 16), using
- OA 8-folding and stacking [i] based on OA(1681, 2097152, S16, 16), using
(65, 81, 1361212)-Net over F16 — Digital
Digital (65, 81, 1361212)-net over F16, using
(65, 81, large)-Net in Base 16 — Upper bound on s
There is no (65, 81, large)-net in base 16, because
- 14 times m-reduction [i] would yield (65, 67, large)-net in base 16, but