Best Known (39−9, 39, s)-Nets in Base 128
(39−9, 39, 2105278)-Net over F128 — Constructive and digital
Digital (30, 39, 2105278)-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 (24, 33, 2097150)-net over F128, using
- net defined by OOA [i] based on linear OOA(12833, 2097150, F128, 9, 9) (dual of [(2097150, 9), 18874317, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(12833, 8388601, F128, 9) (dual of [8388601, 8388568, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(12833, large, F128, 9) (dual of [large, large−33, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 15790321 | 1288−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(12833, large, F128, 9) (dual of [large, large−33, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(12833, 8388601, F128, 9) (dual of [8388601, 8388568, 10]-code), using
- net defined by OOA [i] based on linear OOA(12833, 2097150, F128, 9, 9) (dual of [(2097150, 9), 18874317, 10]-NRT-code), using
- digital (2, 6, 8128)-net over F128, using
(39−9, 39, 3145727)-Net in Base 128 — Constructive
(30, 39, 3145727)-net in base 128, using
- (u, u+v)-construction [i] based on
- digital (6, 10, 1048577)-net over F128, using
- net defined by OOA [i] based on linear OOA(12810, 1048577, F128, 4, 4) (dual of [(1048577, 4), 4194298, 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
- net defined by OOA [i] based on linear OOA(12810, 1048577, F128, 4, 4) (dual of [(1048577, 4), 4194298, 5]-NRT-code), using
- (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
- digital (6, 10, 1048577)-net over F128, using
(39−9, 39, large)-Net over F128 — Digital
Digital (30, 39, large)-net over F128, using
- t-expansion [i] based on digital (29, 39, large)-net over F128, using
- 1 times m-reduction [i] based on digital (29, 40, large)-net over F128, using
(39−9, 39, large)-Net in Base 128 — Upper bound on s
There is no (30, 39, large)-net in base 128, because
- 7 times m-reduction [i] would yield (30, 32, large)-net in base 128, but