Best Known (144−21, 144, s)-Nets in Base 8
(144−21, 144, 209751)-Net over F8 — Constructive and digital
Digital (123, 144, 209751)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (7, 17, 36)-net over F8, using
- digital (106, 127, 209715)-net over F8, using
- net defined by OOA [i] based on linear OOA(8127, 209715, F8, 21, 21) (dual of [(209715, 21), 4403888, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(8127, 2097151, F8, 21) (dual of [2097151, 2097024, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(8127, 2097152, F8, 21) (dual of [2097152, 2097025, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(8127, 2097152, F8, 21) (dual of [2097152, 2097025, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(8127, 2097151, F8, 21) (dual of [2097151, 2097024, 22]-code), using
- net defined by OOA [i] based on linear OOA(8127, 209715, F8, 21, 21) (dual of [(209715, 21), 4403888, 22]-NRT-code), using
(144−21, 144, 419430)-Net in Base 8 — Constructive
(123, 144, 419430)-net in base 8, using
- net defined by OOA [i] based on OOA(8144, 419430, S8, 21, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(8144, 4194301, S8, 21), using
- discarding factors based on OA(8144, 4194310, S8, 21), using
- trace code [i] based on OA(6472, 2097155, S64, 21), using
- discarding parts of the base [i] based on linear OA(12861, 2097155, F128, 21) (dual of [2097155, 2097094, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- linear OA(12861, 2097152, F128, 21) (dual of [2097152, 2097091, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(12858, 2097152, F128, 20) (dual of [2097152, 2097094, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(20) ⊂ Ce(19) [i] based on
- discarding parts of the base [i] based on linear OA(12861, 2097155, F128, 21) (dual of [2097155, 2097094, 22]-code), using
- trace code [i] based on OA(6472, 2097155, S64, 21), using
- discarding factors based on OA(8144, 4194310, S8, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(8144, 4194301, S8, 21), using
(144−21, 144, 3771006)-Net over F8 — Digital
Digital (123, 144, 3771006)-net over F8, using
(144−21, 144, large)-Net in Base 8 — Upper bound on s
There is no (123, 144, large)-net in base 8, because
- 19 times m-reduction [i] would yield (123, 125, large)-net in base 8, but