Best Known (8, 12, s)-Nets in Base 128
(8, 12, 1048706)-Net over F128 — Constructive and digital
Digital (8, 12, 1048706)-net over F128, using
- net defined by OOA [i] based on linear OOA(12812, 1048706, F128, 4, 4) (dual of [(1048706, 4), 4194812, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(12812, 1048706, F128, 3, 4) (dual of [(1048706, 3), 3146106, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(1282, 129, F128, 3, 2) (dual of [(129, 3), 385, 3]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(3;385,128) [i]
- linear OOA(12810, 1048577, F128, 3, 4) (dual of [(1048577, 3), 3145721, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(12810, 2097154, F128, 4) (dual of [2097154, 2097144, 5]-code), using
- discarding factors / shortening the dual code 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 factors / shortening the dual code based on linear OA(12810, 2097155, F128, 4) (dual of [2097155, 2097145, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(12810, 2097154, F128, 4) (dual of [2097154, 2097144, 5]-code), using
- linear OOA(1282, 129, F128, 3, 2) (dual of [(129, 3), 385, 3]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(12812, 1048706, F128, 3, 4) (dual of [(1048706, 3), 3146106, 5]-NRT-code), using
(8, 12, 3840786)-Net over F128 — Digital
Digital (8, 12, 3840786)-net over F128, using
- net defined by OOA [i] based on linear OOA(12812, 3840786, F128, 4, 4) (dual of [(3840786, 4), 15363132, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(12812, 3840786, F128, 3, 4) (dual of [(3840786, 3), 11522346, 5]-NRT-code), using
(8, 12, 4194301)-Net in Base 128 — Constructive
(8, 12, 4194301)-net in base 128, using
- net defined by OOA [i] based on OOA(12812, 4194301, S128, 4, 4), using
- OA 2-folding and stacking [i] based on OA(12812, 8388602, S128, 4), using
- discarding factors based on OA(12812, large, S128, 4), using
- discarding parts of the base [i] based on linear OA(25610, large, F256, 4) (dual of [large, large−10, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding parts of the base [i] based on linear OA(25610, large, F256, 4) (dual of [large, large−10, 5]-code), using
- discarding factors based on OA(12812, large, S128, 4), using
- OA 2-folding and stacking [i] based on OA(12812, 8388602, S128, 4), using
(8, 12, large)-Net in Base 128
(8, 12, large)-net in base 128, using
- net defined by OOA [i] based on OOA(12812, large, S128, 4, 4), using
- appending kth column [i] based on OOA(12812, large, S128, 3, 4), using
- discarding parts of the base [i] based on linear OOA(25610, large, F256, 3, 4), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(25610, large, F256, 4) (dual of [large, large−10, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(25610, large, F256, 4) (dual of [large, large−10, 5]-code), using
- discarding parts of the base [i] based on linear OOA(25610, large, F256, 3, 4), using
- appending kth column [i] based on OOA(12812, large, S128, 3, 4), using
(8, 12, large)-Net in Base 128 — Upper bound on s
There is no (8, 12, large)-net in base 128, because
- 2 times m-reduction [i] would yield (8, 10, large)-net in base 128, but