Best Known (31−9, 31, s)-Nets in Base 128
(31−9, 31, 532416)-Net over F128 — Constructive and digital
Digital (22, 31, 532416)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (2, 6, 8128)-net over F128, using
- net defined by OOA [i] based on linear OOA(1286, 8128, F128, 4, 4) (dual of [(8128, 4), 32506, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(1286, 16256, F128, 4) (dual of [16256, 16250, 5]-code), using
- 1 times truncation [i] based on linear OA(1287, 16257, F128, 5) (dual of [16257, 16250, 6]-code), using
- OA 2-folding and stacking [i] based on linear OA(1286, 16256, F128, 4) (dual of [16256, 16250, 5]-code), using
- net defined by OOA [i] based on linear OOA(1286, 8128, F128, 4, 4) (dual of [(8128, 4), 32506, 5]-NRT-code), using
- digital (16, 25, 524288)-net over F128, using
- net defined by OOA [i] based on linear OOA(12825, 524288, F128, 9, 9) (dual of [(524288, 9), 4718567, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(12825, 2097153, F128, 9) (dual of [2097153, 2097128, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- OOA 4-folding and stacking with additional row [i] based on linear OA(12825, 2097153, F128, 9) (dual of [2097153, 2097128, 10]-code), using
- net defined by OOA [i] based on linear OOA(12825, 524288, F128, 9, 9) (dual of [(524288, 9), 4718567, 10]-NRT-code), using
- digital (2, 6, 8128)-net over F128, using
(31−9, 31, 2097150)-Net in Base 128 — Constructive
(22, 31, 2097150)-net in base 128, using
- 1282 times duplication [i] based on (20, 29, 2097150)-net in base 128, using
- net defined by OOA [i] based on OOA(12829, 2097150, S128, 9, 9), using
- OOA 4-folding and stacking with additional row [i] based on OA(12829, 8388601, S128, 9), using
- discarding factors based on OA(12829, large, S128, 9), using
- discarding parts of the base [i] based on linear OA(25625, large, F256, 9) (dual of [large, large−25, 10]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding parts of the base [i] based on linear OA(25625, large, F256, 9) (dual of [large, large−25, 10]-code), using
- discarding factors based on OA(12829, large, S128, 9), using
- OOA 4-folding and stacking with additional row [i] based on OA(12829, 8388601, S128, 9), using
- net defined by OOA [i] based on OOA(12829, 2097150, S128, 9, 9), using
(31−9, 31, 4338358)-Net over F128 — Digital
Digital (22, 31, 4338358)-net over F128, using
(31−9, 31, 8388602)-Net in Base 128
(22, 31, 8388602)-net in base 128, using
- net defined by OOA [i] based on OOA(12831, 8388602, S128, 12, 9), using
- OOA stacking with additional row [i] based on OOA(12831, large, S128, 4, 9), using
- discarding parts of the base [i] based on linear OOA(25627, large, F256, 4, 9), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(25627, large, F256, 9) (dual of [large, large−27, 10]-code), using
- 2 times code embedding in larger space [i] based on linear OA(25625, large, F256, 9) (dual of [large, large−25, 10]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- 2 times code embedding in larger space [i] based on linear OA(25625, large, F256, 9) (dual of [large, large−25, 10]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(25627, large, F256, 9) (dual of [large, large−27, 10]-code), using
- discarding parts of the base [i] based on linear OOA(25627, large, F256, 4, 9), using
- OOA stacking with additional row [i] based on OOA(12831, large, S128, 4, 9), using
(31−9, 31, large)-Net in Base 128 — Upper bound on s
There is no (22, 31, large)-net in base 128, because
- 7 times m-reduction [i] would yield (22, 24, large)-net in base 128, but