Best Known (24−4, 24, s)-Nets in Base 8
(24−4, 24, 1048588)-Net over F8 — Constructive and digital
Digital (20, 24, 1048588)-net over F8, using
- net defined by OOA [i] based on linear OOA(824, 1048588, F8, 4, 4) (dual of [(1048588, 4), 4194328, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(824, 1048588, F8, 3, 4) (dual of [(1048588, 3), 3145740, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(82, 9, F8, 3, 2) (dual of [(9, 3), 25, 3]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(3;25,8) [i]
- linear OOA(822, 1048579, F8, 3, 4) (dual of [(1048579, 3), 3145715, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(822, 2097158, F8, 4) (dual of [2097158, 2097136, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(822, 2097159, F8, 4) (dual of [2097159, 2097137, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- 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(815, 2097152, F8, 3) (dual of [2097152, 2097137, 4]-code or 2097152-cap in PG(14,8)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [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(3) ⊂ Ce(2) [i] based on
- discarding factors / shortening the dual code based on linear OA(822, 2097159, F8, 4) (dual of [2097159, 2097137, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(822, 2097158, F8, 4) (dual of [2097158, 2097136, 5]-code), using
- linear OOA(82, 9, F8, 3, 2) (dual of [(9, 3), 25, 3]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(824, 1048588, F8, 3, 4) (dual of [(1048588, 3), 3145740, 5]-NRT-code), using
(24−4, 24, 2097155)-Net in Base 8 — Constructive
(20, 24, 2097155)-net in base 8, using
- net defined by OOA [i] based on OOA(824, 2097155, S8, 4, 4), using
- appending kth column [i] based on OOA(824, 2097155, S8, 3, 4), using
- OA 2-folding and stacking [i] based on OA(824, 4194310, S8, 4), using
- trace code [i] based on OA(6412, 2097155, S64, 4), using
- discarding parts of the base [i] based on linear OA(12810, 2097155, F128, 4) (dual of [2097155, 2097145, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(12810, 2097152, F128, 4) (dual of [2097152, 2097142, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(1287, 2097152, F128, 3) (dual of [2097152, 2097145, 4]-code or 2097152-cap in PG(6,128)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- discarding parts of the base [i] based on linear OA(12810, 2097155, F128, 4) (dual of [2097155, 2097145, 5]-code), using
- trace code [i] based on OA(6412, 2097155, S64, 4), using
- OA 2-folding and stacking [i] based on OA(824, 4194310, S8, 4), using
- appending kth column [i] based on OOA(824, 2097155, S8, 3, 4), using
(24−4, 24, 4355176)-Net over F8 — Digital
Digital (20, 24, 4355176)-net over F8, using
- net defined by OOA [i] based on linear OOA(824, 4355176, F8, 4, 4) (dual of [(4355176, 4), 17420680, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(824, 4355176, F8, 3, 4) (dual of [(4355176, 3), 13065504, 5]-NRT-code), using
(24−4, 24, large)-Net in Base 8 — Upper bound on s
There is no (20, 24, large)-net in base 8, because
- 2 times m-reduction [i] would yield (20, 22, large)-net in base 8, but