Best Known (24−7, 24, s)-Nets in Base 128
(24−7, 24, 748207)-Net over F128 — Constructive and digital
Digital (17, 24, 748207)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (2, 5, 49156)-net over F128, using
- net defined by OOA [i] based on linear OOA(1285, 49156, F128, 3, 3) (dual of [(49156, 3), 147463, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(1285, 49156, F128, 2, 3) (dual of [(49156, 2), 98307, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(1285, 49156, F128, 3, 3) (dual of [(49156, 3), 147463, 4]-NRT-code), using
- digital (12, 19, 699051)-net over F128, using
- net defined by OOA [i] based on linear OOA(12819, 699051, F128, 7, 7) (dual of [(699051, 7), 4893338, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(12819, 2097154, F128, 7) (dual of [2097154, 2097135, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(12819, 2097155, F128, 7) (dual of [2097155, 2097136, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(12819, 2097152, F128, 7) (dual of [2097152, 2097133, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(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(6) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(12819, 2097155, F128, 7) (dual of [2097155, 2097136, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(12819, 2097154, F128, 7) (dual of [2097154, 2097135, 8]-code), using
- net defined by OOA [i] based on linear OOA(12819, 699051, F128, 7, 7) (dual of [(699051, 7), 4893338, 8]-NRT-code), using
- digital (2, 5, 49156)-net over F128, using
(24−7, 24, 2796200)-Net in Base 128 — Constructive
(17, 24, 2796200)-net in base 128, using
- base change [i] based on digital (14, 21, 2796200)-net over F256, using
- 2562 times duplication [i] based on digital (12, 19, 2796200)-net over F256, using
- net defined by OOA [i] based on linear OOA(25619, 2796200, F256, 7, 7) (dual of [(2796200, 7), 19573381, 8]-NRT-code), using
- appending kth column [i] based on linear OOA(25619, 2796200, F256, 6, 7) (dual of [(2796200, 6), 16777181, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(25619, 8388601, F256, 7) (dual of [8388601, 8388582, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(25619, large, F256, 7) (dual of [large, large−19, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(25619, large, F256, 7) (dual of [large, large−19, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(25619, 8388601, F256, 7) (dual of [8388601, 8388582, 8]-code), using
- appending kth column [i] based on linear OOA(25619, 2796200, F256, 6, 7) (dual of [(2796200, 6), 16777181, 8]-NRT-code), using
- net defined by OOA [i] based on linear OOA(25619, 2796200, F256, 7, 7) (dual of [(2796200, 7), 19573381, 8]-NRT-code), using
- 2562 times duplication [i] based on digital (12, 19, 2796200)-net over F256, using
(24−7, 24, 6327883)-Net over F128 — Digital
Digital (17, 24, 6327883)-net over F128, using
(24−7, 24, large)-Net in Base 128
(17, 24, large)-net in base 128, using
- base change [i] based on digital (14, 21, large)-net over F256, using
- 2561 times duplication [i] based on digital (13, 20, large)-net over F256, using
- net defined by OOA [i] based on linear OOA(25620, large, F256, 7, 7), using
- appending kth column [i] based on linear OOA(25620, large, F256, 6, 7), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(25620, large, F256, 7) (dual of [large, large−20, 8]-code), using
- 1 times code embedding in larger space [i] based on linear OA(25619, large, F256, 7) (dual of [large, large−19, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- 1 times code embedding in larger space [i] based on linear OA(25619, large, F256, 7) (dual of [large, large−19, 8]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(25620, large, F256, 7) (dual of [large, large−20, 8]-code), using
- appending kth column [i] based on linear OOA(25620, large, F256, 6, 7), using
- net defined by OOA [i] based on linear OOA(25620, large, F256, 7, 7), using
- 2561 times duplication [i] based on digital (13, 20, large)-net over F256, using
(24−7, 24, large)-Net in Base 128 — Upper bound on s
There is no (17, 24, large)-net in base 128, because
- 5 times m-reduction [i] would yield (17, 19, large)-net in base 128, but