Best Known (63, 78, s)-Nets in Base 16
(63, 78, 149814)-Net over F16 — Constructive and digital
Digital (63, 78, 149814)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (0, 7, 17)-net over F16, using
- net from sequence [i] based on digital (0, 16)-sequence over F16, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 0 and N(F) ≥ 17, using
- the rational function field F16(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 16)-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 (0, 7, 17)-net over F16, using
(63, 78, 299594)-Net in Base 16 — Constructive
(63, 78, 299594)-net in base 16, using
- 161 times duplication [i] based on (62, 77, 299594)-net in base 16, using
- base change [i] based on digital (29, 44, 299594)-net over F128, using
- net defined by OOA [i] based on linear OOA(12844, 299594, F128, 15, 15) (dual of [(299594, 15), 4493866, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(12844, 2097159, F128, 15) (dual of [2097159, 2097115, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(12844, 2097160, F128, 15) (dual of [2097160, 2097116, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [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(12837, 2097153, F128, 13) (dual of [2097153, 2097116, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (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([0,7]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12844, 2097160, F128, 15) (dual of [2097160, 2097116, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(12844, 2097159, F128, 15) (dual of [2097159, 2097115, 16]-code), using
- net defined by OOA [i] based on linear OOA(12844, 299594, F128, 15, 15) (dual of [(299594, 15), 4493866, 16]-NRT-code), using
- base change [i] based on digital (29, 44, 299594)-net over F128, using
(63, 78, 2060803)-Net over F16 — Digital
Digital (63, 78, 2060803)-net over F16, using
(63, 78, large)-Net in Base 16 — Upper bound on s
There is no (63, 78, large)-net in base 16, because
- 13 times m-reduction [i] would yield (63, 65, large)-net in base 16, but