Best Known (72−27, 72, s)-Nets in Base 128
(72−27, 72, 1518)-Net over F128 — Constructive and digital
Digital (45, 72, 1518)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (6, 19, 258)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (0, 6, 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, 13, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128 (see above)
- digital (0, 6, 129)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (26, 53, 1260)-net over F128, using
- net defined by OOA [i] based on linear OOA(12853, 1260, F128, 27, 27) (dual of [(1260, 27), 33967, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(12853, 16381, F128, 27) (dual of [16381, 16328, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(12853, 16384, F128, 27) (dual of [16384, 16331, 28]-code), using
- an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- discarding factors / shortening the dual code based on linear OA(12853, 16384, F128, 27) (dual of [16384, 16331, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(12853, 16381, F128, 27) (dual of [16381, 16328, 28]-code), using
- net defined by OOA [i] based on linear OOA(12853, 1260, F128, 27, 27) (dual of [(1260, 27), 33967, 28]-NRT-code), using
- digital (6, 19, 258)-net over F128, using
(72−27, 72, 5043)-Net in Base 128 — Constructive
(45, 72, 5043)-net in base 128, using
- base change [i] based on digital (36, 63, 5043)-net over F256, using
- 2563 times duplication [i] based on digital (33, 60, 5043)-net over F256, using
- net defined by OOA [i] based on linear OOA(25660, 5043, F256, 27, 27) (dual of [(5043, 27), 136101, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(25660, 65560, F256, 27) (dual of [65560, 65500, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,9]) [i] based on
- linear OA(25653, 65537, F256, 27) (dual of [65537, 65484, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(25637, 65537, F256, 19) (dual of [65537, 65500, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(2567, 23, F256, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,256)), using
- discarding factors / shortening the dual code based on linear OA(2567, 256, F256, 7) (dual of [256, 249, 8]-code or 256-arc in PG(6,256)), using
- Reed–Solomon code RS(249,256) [i]
- discarding factors / shortening the dual code based on linear OA(2567, 256, F256, 7) (dual of [256, 249, 8]-code or 256-arc in PG(6,256)), using
- construction X applied to C([0,13]) ⊂ C([0,9]) [i] based on
- OOA 13-folding and stacking with additional row [i] based on linear OA(25660, 65560, F256, 27) (dual of [65560, 65500, 28]-code), using
- net defined by OOA [i] based on linear OOA(25660, 5043, F256, 27, 27) (dual of [(5043, 27), 136101, 28]-NRT-code), using
- 2563 times duplication [i] based on digital (33, 60, 5043)-net over F256, using
(72−27, 72, 56877)-Net over F128 — Digital
Digital (45, 72, 56877)-net over F128, using
(72−27, 72, large)-Net in Base 128 — Upper bound on s
There is no (45, 72, large)-net in base 128, because
- 25 times m-reduction [i] would yield (45, 47, large)-net in base 128, but