Best Known (81−15, 81, s)-Nets in Base 16
(81−15, 81, 149835)-Net over F16 — Constructive and digital
Digital (66, 81, 149835)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (3, 10, 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 (56, 71, 149797)-net over F16, using
- net defined by OOA [i] based on linear OOA(1671, 149797, F16, 15, 15) (dual of [(149797, 15), 2246884, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(1671, 1048580, F16, 15) (dual of [1048580, 1048509, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(1671, 1048581, F16, 15) (dual of [1048581, 1048510, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- linear OA(1671, 1048576, F16, 15) (dual of [1048576, 1048505, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(1666, 1048576, F16, 14) (dual of [1048576, 1048510, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(160, 5, F16, 0) (dual of [5, 5, 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(14) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(1671, 1048581, F16, 15) (dual of [1048581, 1048510, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(1671, 1048580, F16, 15) (dual of [1048580, 1048509, 16]-code), using
- net defined by OOA [i] based on linear OOA(1671, 149797, F16, 15, 15) (dual of [(149797, 15), 2246884, 16]-NRT-code), using
- digital (3, 10, 38)-net over F16, using
(81−15, 81, 299595)-Net in Base 16 — Constructive
(66, 81, 299595)-net in base 16, using
- net defined by OOA [i] based on OOA(1681, 299595, S16, 15, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(1681, 2097166, S16, 15), using
- discarding factors based on OA(1681, 2097168, S16, 15), using
- discarding parts of the base [i] based on linear OA(12846, 2097168, F128, 15) (dual of [2097168, 2097122, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- 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(12831, 2097153, F128, 11) (dual of [2097153, 2097122, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(1283, 15, F128, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,128) or 15-cap in PG(2,128)), using
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- Reed–Solomon code RS(125,128) [i]
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- discarding parts of the base [i] based on linear OA(12846, 2097168, F128, 15) (dual of [2097168, 2097122, 16]-code), using
- discarding factors based on OA(1681, 2097168, S16, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(1681, 2097166, S16, 15), using
(81−15, 81, 3733030)-Net over F16 — Digital
Digital (66, 81, 3733030)-net over F16, using
(81−15, 81, large)-Net in Base 16 — Upper bound on s
There is no (66, 81, large)-net in base 16, because
- 13 times m-reduction [i] would yield (66, 68, large)-net in base 16, but