Best Known (72−14, 72, s)-Nets in Base 16
(72−14, 72, 149801)-Net over F16 — Constructive and digital
Digital (58, 72, 149801)-net over F16, using
- net defined by OOA [i] based on linear OOA(1672, 149801, F16, 14, 14) (dual of [(149801, 14), 2097142, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(1672, 1048607, F16, 14) (dual of [1048607, 1048535, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(8) [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(1641, 1048576, F16, 9) (dual of [1048576, 1048535, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(166, 31, F16, 4) (dual of [31, 25, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(166, 240, F16, 4) (dual of [240, 234, 5]-code), using
- 1 times truncation [i] based on linear OA(167, 241, F16, 5) (dual of [241, 234, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(166, 240, F16, 4) (dual of [240, 234, 5]-code), using
- construction X applied to Ce(13) ⊂ Ce(8) [i] based on
- OA 7-folding and stacking [i] based on linear OA(1672, 1048607, F16, 14) (dual of [1048607, 1048535, 15]-code), using
(72−14, 72, 299594)-Net in Base 16 — Constructive
(58, 72, 299594)-net in base 16, using
- net defined by OOA [i] based on OOA(1672, 299594, S16, 14, 14), using
- OA 7-folding and stacking [i] based on OA(1672, 2097158, S16, 14), using
- discarding factors based on OA(1672, 2097159, S16, 14), using
- discarding parts of the base [i] 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 parts of the base [i] based on linear OA(12841, 2097159, F128, 14) (dual of [2097159, 2097118, 15]-code), using
- discarding factors based on OA(1672, 2097159, S16, 14), using
- OA 7-folding and stacking [i] based on OA(1672, 2097158, S16, 14), using
(72−14, 72, 1763148)-Net over F16 — Digital
Digital (58, 72, 1763148)-net over F16, using
(72−14, 72, large)-Net in Base 16 — Upper bound on s
There is no (58, 72, large)-net in base 16, because
- 12 times m-reduction [i] would yield (58, 60, large)-net in base 16, but