Best Known (72−25, 72, s)-Nets in Base 128
(72−25, 72, 2730)-Net over F128 — Constructive and digital
Digital (47, 72, 2730)-net over F128, using
- net defined by OOA [i] based on linear OOA(12872, 2730, F128, 25, 25) (dual of [(2730, 25), 68178, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(12872, 32761, F128, 25) (dual of [32761, 32689, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(12872, 32769, F128, 25) (dual of [32769, 32697, 26]-code), using
- (u, u+v)-construction [i] based on
- linear OA(12823, 16384, F128, 12) (dual of [16384, 16361, 13]-code), using
- an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(12849, 16385, F128, 25) (dual of [16385, 16336, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(12823, 16384, F128, 12) (dual of [16384, 16361, 13]-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(12872, 32769, F128, 25) (dual of [32769, 32697, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(12872, 32761, F128, 25) (dual of [32761, 32689, 26]-code), using
(72−25, 72, 5720)-Net in Base 128 — Constructive
(47, 72, 5720)-net in base 128, using
- base change [i] based on digital (38, 63, 5720)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (2, 14, 259)-net over F256, using
- net from sequence [i] based on digital (2, 258)-sequence over F256, using
- digital (24, 49, 5461)-net over F256, using
- net defined by OOA [i] based on linear OOA(25649, 5461, F256, 25, 25) (dual of [(5461, 25), 136476, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(25649, 65533, F256, 25) (dual of [65533, 65484, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(25649, 65536, F256, 25) (dual of [65536, 65487, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(25649, 65536, F256, 25) (dual of [65536, 65487, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(25649, 65533, F256, 25) (dual of [65533, 65484, 26]-code), using
- net defined by OOA [i] based on linear OOA(25649, 5461, F256, 25, 25) (dual of [(5461, 25), 136476, 26]-NRT-code), using
- digital (2, 14, 259)-net over F256, using
- (u, u+v)-construction [i] based on
(72−25, 72, 161890)-Net over F128 — Digital
Digital (47, 72, 161890)-net over F128, using
(72−25, 72, large)-Net in Base 128 — Upper bound on s
There is no (47, 72, large)-net in base 128, because
- 23 times m-reduction [i] would yield (47, 49, large)-net in base 128, but