Best Known (72, 72+17, s)-Nets in Base 16
(72, 72+17, 131120)-Net over F16 — Constructive and digital
Digital (72, 89, 131120)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (5, 13, 49)-net over F16, using
- net from sequence [i] based on digital (5, 48)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 5 and N(F) ≥ 49, using
- net from sequence [i] based on digital (5, 48)-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 (5, 13, 49)-net over F16, using
(72, 72+17, 262145)-Net in Base 16 — Constructive
(72, 89, 262145)-net in base 16, using
- net defined by OOA [i] based on OOA(1689, 262145, S16, 17, 17), using
- OOA 8-folding and stacking with additional row [i] based on OA(1689, 2097161, S16, 17), using
- 1 times code embedding in larger space [i] based on OA(1688, 2097160, S16, 17), using
- discarding parts of the base [i] based on linear OA(12850, 2097160, F128, 17) (dual of [2097160, 2097110, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- linear OA(12849, 2097153, F128, 17) (dual of [2097153, 2097104, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(12843, 2097153, F128, 15) (dual of [2097153, 2097110, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(1281, 7, F128, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- discarding parts of the base [i] based on linear OA(12850, 2097160, F128, 17) (dual of [2097160, 2097110, 18]-code), using
- 1 times code embedding in larger space [i] based on OA(1688, 2097160, S16, 17), using
- OOA 8-folding and stacking with additional row [i] based on OA(1689, 2097161, S16, 17), using
(72, 72+17, 2261343)-Net over F16 — Digital
Digital (72, 89, 2261343)-net over F16, using
(72, 72+17, large)-Net in Base 16 — Upper bound on s
There is no (72, 89, large)-net in base 16, because
- 15 times m-reduction [i] would yield (72, 74, large)-net in base 16, but