Best Known (54, 54+20, s)-Nets in Base 128
(54, 54+20, 209994)-Net over F128 — Constructive and digital
Digital (54, 74, 209994)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (6, 16, 279)-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 (1, 11, 150)-net over F128, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 1 and N(F) ≥ 150, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- digital (0, 5, 129)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (38, 58, 209715)-net over F128, using
- net defined by OOA [i] based on linear OOA(12858, 209715, F128, 20, 20) (dual of [(209715, 20), 4194242, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(12858, 2097150, F128, 20) (dual of [2097150, 2097092, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(12858, 2097152, F128, 20) (dual of [2097152, 2097094, 21]-code), using
- an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(12858, 2097152, F128, 20) (dual of [2097152, 2097094, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(12858, 2097150, F128, 20) (dual of [2097150, 2097092, 21]-code), using
- net defined by OOA [i] based on linear OOA(12858, 209715, F128, 20, 20) (dual of [(209715, 20), 4194242, 21]-NRT-code), using
- digital (6, 16, 279)-net over F128, using
(54, 54+20, 838860)-Net in Base 128 — Constructive
(54, 74, 838860)-net in base 128, using
- 1282 times duplication [i] based on (52, 72, 838860)-net in base 128, using
- t-expansion [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
- t-expansion [i] based on (51, 72, 838860)-net in base 128, using
(54, 54+20, large)-Net over F128 — Digital
Digital (54, 74, large)-net over F128, using
(54, 54+20, large)-Net in Base 128 — Upper bound on s
There is no (54, 74, large)-net in base 128, because
- 18 times m-reduction [i] would yield (54, 56, large)-net in base 128, but