Best Known (76−21, 76, s)-Nets in Base 128
(76−21, 76, 209973)-Net over F128 — Constructive and digital
Digital (55, 76, 209973)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (5, 15, 258)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (0, 5, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- digital (0, 10, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128 (see above)
- digital (0, 5, 129)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (40, 61, 209715)-net over F128, using
- net defined by OOA [i] based on linear OOA(12861, 209715, F128, 21, 21) (dual of [(209715, 21), 4403954, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(12861, 2097151, F128, 21) (dual of [2097151, 2097090, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(12861, 2097152, F128, 21) (dual of [2097152, 2097091, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(12861, 2097152, F128, 21) (dual of [2097152, 2097091, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(12861, 2097151, F128, 21) (dual of [2097151, 2097090, 22]-code), using
- net defined by OOA [i] based on linear OOA(12861, 209715, F128, 21, 21) (dual of [(209715, 21), 4403954, 22]-NRT-code), using
- digital (5, 15, 258)-net over F128, using
(76−21, 76, 838860)-Net in Base 128 — Constructive
(55, 76, 838860)-net in base 128, using
- 1284 times duplication [i] based on (51, 72, 838860)-net in base 128, using
- base change [i] based on digital (42, 63, 838860)-net over F256, using
- 2562 times duplication [i] based on digital (40, 61, 838860)-net over F256, using
- net defined by OOA [i] based on linear OOA(25661, 838860, F256, 21, 21) (dual of [(838860, 21), 17615999, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(25661, 8388601, F256, 21) (dual of [8388601, 8388540, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(25661, large, F256, 21) (dual of [large, large−61, 22]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- discarding factors / shortening the dual code based on linear OA(25661, large, F256, 21) (dual of [large, large−61, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(25661, 8388601, F256, 21) (dual of [8388601, 8388540, 22]-code), using
- net defined by OOA [i] based on linear OOA(25661, 838860, F256, 21, 21) (dual of [(838860, 21), 17615999, 22]-NRT-code), using
- 2562 times duplication [i] based on digital (40, 61, 838860)-net over F256, using
- base change [i] based on digital (42, 63, 838860)-net over F256, using
(76−21, 76, 6651216)-Net over F128 — Digital
Digital (55, 76, 6651216)-net over F128, using
(76−21, 76, large)-Net in Base 128 — Upper bound on s
There is no (55, 76, large)-net in base 128, because
- 19 times m-reduction [i] would yield (55, 57, large)-net in base 128, but