Best Known (43, 69, s)-Nets in Base 128
(43, 69, 1476)-Net over F128 — Constructive and digital
Digital (43, 69, 1476)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (5, 18, 216)-net over F128, using
- net from sequence [i] based on digital (5, 215)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 5 and N(F) ≥ 216, using
- net from sequence [i] based on digital (5, 215)-sequence over F128, using
- digital (25, 51, 1260)-net over F128, using
- net defined by OOA [i] based on linear OOA(12851, 1260, F128, 26, 26) (dual of [(1260, 26), 32709, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(12851, 16380, F128, 26) (dual of [16380, 16329, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(12851, 16384, F128, 26) (dual of [16384, 16333, 27]-code), using
- an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- discarding factors / shortening the dual code based on linear OA(12851, 16384, F128, 26) (dual of [16384, 16333, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(12851, 16380, F128, 26) (dual of [16380, 16329, 27]-code), using
- net defined by OOA [i] based on linear OOA(12851, 1260, F128, 26, 26) (dual of [(1260, 26), 32709, 27]-NRT-code), using
- digital (5, 18, 216)-net over F128, using
(43, 69, 5043)-Net in Base 128 — Constructive
(43, 69, 5043)-net in base 128, using
- t-expansion [i] based on (42, 69, 5043)-net in base 128, using
- net defined by OOA [i] based on OOA(12869, 5043, S128, 27, 27), using
- OOA 13-folding and stacking with additional row [i] based on OA(12869, 65560, S128, 27), using
- discarding parts of the base [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
- discarding parts of the base [i] based on linear OA(25660, 65560, F256, 27) (dual of [65560, 65500, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on OA(12869, 65560, S128, 27), using
- net defined by OOA [i] based on OOA(12869, 5043, S128, 27, 27), using
(43, 69, 52460)-Net over F128 — Digital
Digital (43, 69, 52460)-net over F128, using
(43, 69, large)-Net in Base 128 — Upper bound on s
There is no (43, 69, large)-net in base 128, because
- 24 times m-reduction [i] would yield (43, 45, large)-net in base 128, but