Best Known (86−16, 86, s)-Nets in Base 16
(86−16, 86, 131109)-Net over F16 — Constructive and digital
Digital (70, 86, 131109)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (3, 11, 38)-net over F16, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 3 and N(F) ≥ 38, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- digital (59, 75, 131071)-net over F16, using
- net defined by OOA [i] based on linear OOA(1675, 131071, F16, 16, 16) (dual of [(131071, 16), 2097061, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(1675, 1048568, F16, 16) (dual of [1048568, 1048493, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(1675, 1048575, F16, 16) (dual of [1048575, 1048500, 17]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(1676, 1048576, F16, 17) (dual of [1048576, 1048500, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(1675, 1048575, F16, 16) (dual of [1048575, 1048500, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(1675, 1048568, F16, 16) (dual of [1048568, 1048493, 17]-code), using
- net defined by OOA [i] based on linear OOA(1675, 131071, F16, 16, 16) (dual of [(131071, 16), 2097061, 17]-NRT-code), using
- digital (3, 11, 38)-net over F16, using
(86−16, 86, 262145)-Net in Base 16 — Constructive
(70, 86, 262145)-net in base 16, using
- 162 times duplication [i] based on (68, 84, 262145)-net in base 16, using
- base change [i] based on digital (32, 48, 262145)-net over F128, using
- net defined by OOA [i] based on linear OOA(12848, 262145, F128, 16, 16) (dual of [(262145, 16), 4194272, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(12848, 2097160, F128, 16) (dual of [2097160, 2097112, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(12848, 2097163, F128, 16) (dual of [2097163, 2097115, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [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(12837, 2097152, F128, 13) (dual of [2097152, 2097115, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(1282, 11, F128, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,128)), using
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- Reed–Solomon code RS(126,128) [i]
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(12848, 2097163, F128, 16) (dual of [2097163, 2097115, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(12848, 2097160, F128, 16) (dual of [2097160, 2097112, 17]-code), using
- net defined by OOA [i] based on linear OOA(12848, 262145, F128, 16, 16) (dual of [(262145, 16), 4194272, 17]-NRT-code), using
- base change [i] based on digital (32, 48, 262145)-net over F128, using
(86−16, 86, 3430027)-Net over F16 — Digital
Digital (70, 86, 3430027)-net over F16, using
(86−16, 86, large)-Net in Base 16 — Upper bound on s
There is no (70, 86, large)-net in base 16, because
- 14 times m-reduction [i] would yield (70, 72, large)-net in base 16, but