Best Known (87−17, 87, s)-Nets in Base 16
(87−17, 87, 131109)-Net over F16 — Constructive and digital
Digital (70, 87, 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, 76, 131071)-net over F16, using
- net defined by OOA [i] based on linear OOA(1676, 131071, F16, 17, 17) (dual of [(131071, 17), 2228131, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(1676, 1048569, F16, 17) (dual of [1048569, 1048493, 18]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(1676, 1048576, F16, 17) (dual of [1048576, 1048500, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(1676, 1048569, F16, 17) (dual of [1048569, 1048493, 18]-code), using
- net defined by OOA [i] based on linear OOA(1676, 131071, F16, 17, 17) (dual of [(131071, 17), 2228131, 18]-NRT-code), using
- digital (3, 11, 38)-net over F16, using
(87−17, 87, 262144)-Net in Base 16 — Constructive
(70, 87, 262144)-net in base 16, using
- 161 times duplication [i] based on (69, 86, 262144)-net in base 16, using
- net defined by OOA [i] based on OOA(1686, 262144, S16, 17, 17), using
- OOA 8-folding and stacking with additional row [i] based on OA(1686, 2097153, S16, 17), using
- discarding factors based on OA(1686, 2097155, S16, 17), using
- discarding parts of the base [i] based on linear OA(12849, 2097155, F128, 17) (dual of [2097155, 2097106, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(12849, 2097152, F128, 17) (dual of [2097152, 2097103, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- 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(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(16) ⊂ Ce(15) [i] based on
- discarding parts of the base [i] based on linear OA(12849, 2097155, F128, 17) (dual of [2097155, 2097106, 18]-code), using
- discarding factors based on OA(1686, 2097155, S16, 17), using
- OOA 8-folding and stacking with additional row [i] based on OA(1686, 2097153, S16, 17), using
- net defined by OOA [i] based on OOA(1686, 262144, S16, 17, 17), using
(87−17, 87, 1599013)-Net over F16 — Digital
Digital (70, 87, 1599013)-net over F16, using
(87−17, 87, large)-Net in Base 16 — Upper bound on s
There is no (70, 87, large)-net in base 16, because
- 15 times m-reduction [i] would yield (70, 72, large)-net in base 16, but