Best Known (122−7, 122, s)-Nets in Base 8
(122−7, 122, large)-Net over F8 — Constructive and digital
Digital (115, 122, large)-net over F8, using
- 3 times m-reduction [i] based on digital (115, 125, large)-net over F8, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 1, 1048579)-net over F8, using
- s-reduction based on digital (0, 1, s)-net over F8 with arbitrarily large s, using
- digital (0, 1, 1048579)-net over F8 (see above)
- digital (0, 1, 1048579)-net over F8 (see above)
- digital (6, 8, 1048579)-net over F8, using
- s-reduction based on digital (6, 8, 2396745)-net over F8, using
- digital (6, 8, 1048579)-net over F8 (see above)
- digital (9, 12, 1048579)-net over F8, using
- s-reduction based on digital (9, 12, 2931457)-net over F8, using
- net defined by OOA [i] based on linear OOA(812, 2931457, F8, 3, 3) (dual of [(2931457, 3), 8794359, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(812, 2931457, F8, 2, 3) (dual of [(2931457, 2), 5862902, 4]-NRT-code), using
- OAs with strength 3, b ≠ 2, and m > 3 are always embeddable [i] based on linear OA(812, 2931457, F8, 3) (dual of [2931457, 2931445, 4]-code or 2931457-cap in PG(11,8)), using
- appending kth column [i] based on linear OOA(812, 2931457, F8, 2, 3) (dual of [(2931457, 2), 5862902, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(812, 2931457, F8, 3, 3) (dual of [(2931457, 3), 8794359, 4]-NRT-code), using
- s-reduction based on digital (9, 12, 2931457)-net over F8, using
- digital (24, 29, 1048579)-net over F8, using
- net defined by OOA [i] based on linear OOA(829, 1048579, F8, 5, 5) (dual of [(1048579, 5), 5242866, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(829, 2097159, F8, 5) (dual of [2097159, 2097130, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(829, 2097152, F8, 5) (dual of [2097152, 2097123, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(822, 2097152, F8, 4) (dual of [2097152, 2097130, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(80, 7, F8, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(829, 2097159, F8, 5) (dual of [2097159, 2097130, 6]-code), using
- net defined by OOA [i] based on linear OOA(829, 1048579, F8, 5, 5) (dual of [(1048579, 5), 5242866, 6]-NRT-code), using
- digital (55, 65, 1677720)-net over F8, using
- net defined by OOA [i] based on linear OOA(865, 1677720, F8, 10, 10) (dual of [(1677720, 10), 16777135, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(865, 8388600, F8, 10) (dual of [8388600, 8388535, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(865, large, F8, 10) (dual of [large, large−65, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(865, large, F8, 10) (dual of [large, large−65, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(865, 8388600, F8, 10) (dual of [8388600, 8388535, 11]-code), using
- net defined by OOA [i] based on linear OOA(865, 1677720, F8, 10, 10) (dual of [(1677720, 10), 16777135, 11]-NRT-code), using
- digital (0, 1, 1048579)-net over F8, using
- generalized (u, u+v)-construction [i] based on
(122−7, 122, large)-Net in Base 8 — Upper bound on s
There is no (115, 122, large)-net in base 8, because
- 5 times m-reduction [i] would yield (115, 117, large)-net in base 8, but