Best Known (47, 47+8, s)-Nets in Base 8
(47, 47+8, 524315)-Net over F8 — Constructive and digital
Digital (47, 55, 524315)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (2, 6, 28)-net over F8, using
- net defined by OOA [i] based on linear OOA(86, 28, F8, 4, 4) (dual of [(28, 4), 106, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(86, 28, F8, 3, 4) (dual of [(28, 3), 78, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(86, 56, F8, 4) (dual of [56, 50, 5]-code), using
- 1 times truncation [i] based on linear OA(87, 57, F8, 5) (dual of [57, 50, 6]-code), using
- OA 2-folding and stacking [i] based on linear OA(86, 56, F8, 4) (dual of [56, 50, 5]-code), using
- appending kth column [i] based on linear OOA(86, 28, F8, 3, 4) (dual of [(28, 3), 78, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(86, 28, F8, 4, 4) (dual of [(28, 4), 106, 5]-NRT-code), using
- digital (41, 49, 524287)-net over F8, using
- net defined by OOA [i] based on linear OOA(849, 524287, F8, 8, 8) (dual of [(524287, 8), 4194247, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(849, 2097148, F8, 8) (dual of [2097148, 2097099, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(849, 2097151, F8, 8) (dual of [2097151, 2097102, 9]-code), using
- the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(849, 2097151, F8, 8) (dual of [2097151, 2097102, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(849, 2097148, F8, 8) (dual of [2097148, 2097099, 9]-code), using
- net defined by OOA [i] based on linear OOA(849, 524287, F8, 8, 8) (dual of [(524287, 8), 4194247, 9]-NRT-code), using
- digital (2, 6, 28)-net over F8, using
(47, 47+8, 1048579)-Net in Base 8 — Constructive
(47, 55, 1048579)-net in base 8, using
- 81 times duplication [i] based on (46, 54, 1048579)-net in base 8, using
- net defined by OOA [i] based on OOA(854, 1048579, S8, 8, 8), using
- OA 4-folding and stacking [i] based on OA(854, 4194316, S8, 8), using
- discarding factors based on OA(854, 4194318, S8, 8), using
- trace code [i] based on OA(6427, 2097159, S64, 8), using
- discarding parts of the base [i] based on linear OA(12823, 2097159, F128, 8) (dual of [2097159, 2097136, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(12822, 2097152, F128, 8) (dual of [2097152, 2097130, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(12816, 2097152, F128, 6) (dual of [2097152, 2097136, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(7) ⊂ Ce(5) [i] based on
- discarding parts of the base [i] based on linear OA(12823, 2097159, F128, 8) (dual of [2097159, 2097136, 9]-code), using
- trace code [i] based on OA(6427, 2097159, S64, 8), using
- discarding factors based on OA(854, 4194318, S8, 8), using
- OA 4-folding and stacking [i] based on OA(854, 4194316, S8, 8), using
- net defined by OOA [i] based on OOA(854, 1048579, S8, 8, 8), using
(47, 47+8, 6019049)-Net over F8 — Digital
Digital (47, 55, 6019049)-net over F8, using
(47, 47+8, large)-Net in Base 8 — Upper bound on s
There is no (47, 55, large)-net in base 8, because
- 6 times m-reduction [i] would yield (47, 49, large)-net in base 8, but