Best Known (39−11, 39, s)-Nets in Base 128
(39−11, 39, 427559)-Net over F128 — Constructive and digital
Digital (28, 39, 427559)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (2, 7, 8128)-net over F128, using
- net defined by OOA [i] based on linear OOA(1287, 8128, F128, 5, 5) (dual of [(8128, 5), 40633, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(1287, 8128, F128, 4, 5) (dual of [(8128, 4), 32505, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(1287, 16257, F128, 5) (dual of [16257, 16250, 6]-code), using
- appending kth column [i] based on linear OOA(1287, 8128, F128, 4, 5) (dual of [(8128, 4), 32505, 6]-NRT-code), using
- net defined by OOA [i] based on linear OOA(1287, 8128, F128, 5, 5) (dual of [(8128, 5), 40633, 6]-NRT-code), using
- digital (21, 32, 419431)-net over F128, using
- net defined by OOA [i] based on linear OOA(12832, 419431, F128, 11, 11) (dual of [(419431, 11), 4613709, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(12832, 2097156, F128, 11) (dual of [2097156, 2097124, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(12832, 2097160, F128, 11) (dual of [2097160, 2097128, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- linear OA(12831, 2097153, F128, 11) (dual of [2097153, 2097122, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- 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]
- linear OA(1281, 7, F128, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12832, 2097160, F128, 11) (dual of [2097160, 2097128, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(12832, 2097156, F128, 11) (dual of [2097156, 2097124, 12]-code), using
- net defined by OOA [i] based on linear OOA(12832, 419431, F128, 11, 11) (dual of [(419431, 11), 4613709, 12]-NRT-code), using
- digital (2, 7, 8128)-net over F128, using
(39−11, 39, 1677720)-Net in Base 128 — Constructive
(28, 39, 1677720)-net in base 128, using
- 1283 times duplication [i] based on (25, 36, 1677720)-net in base 128, using
- net defined by OOA [i] based on OOA(12836, 1677720, S128, 11, 11), using
- OOA 5-folding and stacking with additional row [i] based on OA(12836, 8388601, S128, 11), using
- discarding factors based on OA(12836, large, S128, 11), using
- discarding parts of the base [i] based on linear OA(25631, large, F256, 11) (dual of [large, large−31, 12]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- discarding parts of the base [i] based on linear OA(25631, large, F256, 11) (dual of [large, large−31, 12]-code), using
- discarding factors based on OA(12836, large, S128, 11), using
- OOA 5-folding and stacking with additional row [i] based on OA(12836, 8388601, S128, 11), using
- net defined by OOA [i] based on OOA(12836, 1677720, S128, 11, 11), using
(39−11, 39, 5892395)-Net over F128 — Digital
Digital (28, 39, 5892395)-net over F128, using
(39−11, 39, large)-Net in Base 128 — Upper bound on s
There is no (28, 39, large)-net in base 128, because
- 9 times m-reduction [i] would yield (28, 30, large)-net in base 128, but