Best Known (16, 16+7, s)-Nets in Base 128
(16, 16+7, 715436)-Net over F128 — Constructive and digital
Digital (16, 23, 715436)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (1, 4, 16385)-net over F128, using
- net defined by OOA [i] based on linear OOA(1284, 16385, F128, 3, 3) (dual of [(16385, 3), 49151, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(1284, 16385, F128, 2, 3) (dual of [(16385, 2), 32766, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(1284, 16385, F128, 3, 3) (dual of [(16385, 3), 49151, 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 (1, 4, 16385)-net over F128, using
(16, 16+7, 2796200)-Net in Base 128 — Constructive
(16, 23, 2796200)-net in base 128, using
- 1281 times duplication [i] based on (15, 22, 2796200)-net in base 128, using
- net defined by OOA [i] based on OOA(12822, 2796200, S128, 7, 7), using
- OOA 3-folding and stacking with additional row [i] based on OA(12822, 8388601, S128, 7), using
- discarding factors based on OA(12822, large, S128, 7), using
- discarding parts of the base [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]
- discarding parts of the base [i] based on linear OA(25619, large, F256, 7) (dual of [large, large−19, 8]-code), using
- discarding factors based on OA(12822, large, S128, 7), using
- OOA 3-folding and stacking with additional row [i] based on OA(12822, 8388601, S128, 7), using
- net defined by OOA [i] based on OOA(12822, 2796200, S128, 7, 7), using
(16, 16+7, 2818753)-Net over F128 — Digital
Digital (16, 23, 2818753)-net over F128, using
(16, 16+7, 8388602)-Net in Base 128
(16, 23, 8388602)-net in base 128, using
- net defined by OOA [i] based on OOA(12823, 8388602, S128, 9, 7), using
- OOA stacking with additional row [i] based on OOA(12823, large, S128, 3, 7), using
- discarding parts of the base [i] based on linear OOA(25620, large, F256, 3, 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
- discarding parts of the base [i] based on linear OOA(25620, large, F256, 3, 7), using
- OOA stacking with additional row [i] based on OOA(12823, large, S128, 3, 7), using
(16, 16+7, large)-Net in Base 128 — Upper bound on s
There is no (16, 23, large)-net in base 128, because
- 5 times m-reduction [i] would yield (16, 18, large)-net in base 128, but