Best Known (75−14, 75, s)-Nets in Base 16
(75−14, 75, 149830)-Net over F16 — Constructive and digital
Digital (61, 75, 149830)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (2, 9, 33)-net over F16, using
- net from sequence [i] based on digital (2, 32)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 2 and N(F) ≥ 33, using
- net from sequence [i] based on digital (2, 32)-sequence over F16, using
- digital (52, 66, 149797)-net over F16, using
- net defined by OOA [i] based on linear OOA(1666, 149797, F16, 14, 14) (dual of [(149797, 14), 2097092, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(1666, 1048579, F16, 14) (dual of [1048579, 1048513, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(1666, 1048581, F16, 14) (dual of [1048581, 1048515, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- 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(1661, 1048576, F16, 13) (dual of [1048576, 1048515, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(1666, 1048581, F16, 14) (dual of [1048581, 1048515, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(1666, 1048579, F16, 14) (dual of [1048579, 1048513, 15]-code), using
- net defined by OOA [i] based on linear OOA(1666, 149797, F16, 14, 14) (dual of [(149797, 14), 2097092, 15]-NRT-code), using
- digital (2, 9, 33)-net over F16, using
(75−14, 75, 299594)-Net in Base 16 — Constructive
(61, 75, 299594)-net in base 16, using
- base change [i] based on (36, 50, 299594)-net in base 64, using
- 641 times duplication [i] based on (35, 49, 299594)-net in base 64, using
- base change [i] based on digital (28, 42, 299594)-net over F128, using
- 1281 times duplication [i] based on digital (27, 41, 299594)-net over F128, using
- net defined by OOA [i] based on linear OOA(12841, 299594, F128, 14, 14) (dual of [(299594, 14), 4194275, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(12841, 2097158, F128, 14) (dual of [2097158, 2097117, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(12841, 2097159, F128, 14) (dual of [2097159, 2097118, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- linear OA(12840, 2097152, F128, 14) (dual of [2097152, 2097112, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(12834, 2097152, F128, 12) (dual of [2097152, 2097118, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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 Ce(13) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(12841, 2097159, F128, 14) (dual of [2097159, 2097118, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(12841, 2097158, F128, 14) (dual of [2097158, 2097117, 15]-code), using
- net defined by OOA [i] based on linear OOA(12841, 299594, F128, 14, 14) (dual of [(299594, 14), 4194275, 15]-NRT-code), using
- 1281 times duplication [i] based on digital (27, 41, 299594)-net over F128, using
- base change [i] based on digital (28, 42, 299594)-net over F128, using
- 641 times duplication [i] based on (35, 49, 299594)-net in base 64, using
(75−14, 75, 3343196)-Net over F16 — Digital
Digital (61, 75, 3343196)-net over F16, using
(75−14, 75, large)-Net in Base 16 — Upper bound on s
There is no (61, 75, large)-net in base 16, because
- 12 times m-reduction [i] would yield (61, 63, large)-net in base 16, but