Best Known (27, 27+9, s)-Nets in Base 128
(27, 27+9, 2097150)-Net over F128 — Constructive and digital
Digital (27, 36, 2097150)-net over F128, using
- 1283 times duplication [i] based on 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
(27, 27+9, 2129790)-Net in Base 128 — Constructive
(27, 36, 2129790)-net in base 128, using
- (u, u+v)-construction [i] based on
- (3, 7, 32640)-net in base 128, using
- 1 times m-reduction [i] based on (3, 8, 32640)-net in base 128, using
- base change [i] based on digital (2, 7, 32640)-net over F256, using
- net defined by OOA [i] based on linear OOA(2567, 32640, F256, 5, 5) (dual of [(32640, 5), 163193, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(2567, 65281, F256, 5) (dual of [65281, 65274, 6]-code), using
- net defined by OOA [i] based on linear OOA(2567, 32640, F256, 5, 5) (dual of [(32640, 5), 163193, 6]-NRT-code), using
- base change [i] based on digital (2, 7, 32640)-net over F256, using
- 1 times m-reduction [i] based on (3, 8, 32640)-net in base 128, 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
- (3, 7, 32640)-net in base 128, using
(27, 27+9, large)-Net over F128 — Digital
Digital (27, 36, large)-net over F128, using
- t-expansion [i] based on digital (26, 36, large)-net over F128, using
(27, 27+9, large)-Net in Base 128 — Upper bound on s
There is no (27, 36, large)-net in base 128, because
- 7 times m-reduction [i] would yield (27, 29, large)-net in base 128, but