Best Known (58−13, 58, s)-Nets in Base 128
(58−13, 58, 1398358)-Net over F128 — Constructive and digital
Digital (45, 58, 1398358)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (3, 9, 258)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (0, 3, 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, 6, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128 (see above)
- digital (0, 3, 129)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (36, 49, 1398100)-net over F128, using
- net defined by OOA [i] based on linear OOA(12849, 1398100, F128, 13, 13) (dual of [(1398100, 13), 18175251, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(12849, 8388601, F128, 13) (dual of [8388601, 8388552, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(12849, large, F128, 13) (dual of [large, large−49, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 15790321 | 1288−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(12849, large, F128, 13) (dual of [large, large−49, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(12849, 8388601, F128, 13) (dual of [8388601, 8388552, 14]-code), using
- net defined by OOA [i] based on linear OOA(12849, 1398100, F128, 13, 13) (dual of [(1398100, 13), 18175251, 14]-NRT-code), using
- digital (3, 9, 258)-net over F128, using
(58−13, 58, 1419948)-Net in Base 128 — Constructive
(45, 58, 1419948)-net in base 128, using
- (u, u+v)-construction [i] based on
- (9, 15, 21848)-net in base 128, using
- net defined by OOA [i] based on OOA(12815, 21848, S128, 6, 6), using
- OA 3-folding and stacking [i] based on OA(12815, 65544, S128, 6), using
- discarding parts of the base [i] based on linear OA(25613, 65544, F256, 6) (dual of [65544, 65531, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- linear OA(25611, 65536, F256, 6) (dual of [65536, 65525, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(2565, 65536, F256, 3) (dual of [65536, 65531, 4]-code or 65536-cap in PG(4,256)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(2562, 8, F256, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,256)), using
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- Reed–Solomon code RS(254,256) [i]
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- discarding parts of the base [i] based on linear OA(25613, 65544, F256, 6) (dual of [65544, 65531, 7]-code), using
- OA 3-folding and stacking [i] based on OA(12815, 65544, S128, 6), using
- net defined by OOA [i] based on OOA(12815, 21848, S128, 6, 6), using
- (30, 43, 1398100)-net in base 128, using
- net defined by OOA [i] based on OOA(12843, 1398100, S128, 13, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(12843, 8388601, S128, 13), using
- discarding factors based on OA(12843, large, S128, 13), using
- discarding parts of the base [i] based on linear OA(25637, large, F256, 13) (dual of [large, large−37, 14]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- discarding parts of the base [i] based on linear OA(25637, large, F256, 13) (dual of [large, large−37, 14]-code), using
- discarding factors based on OA(12843, large, S128, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(12843, 8388601, S128, 13), using
- net defined by OOA [i] based on OOA(12843, 1398100, S128, 13, 13), using
- (9, 15, 21848)-net in base 128, using
(58−13, 58, large)-Net over F128 — Digital
Digital (45, 58, large)-net over F128, using
- t-expansion [i] based on digital (43, 58, large)-net over F128, using
- 1 times m-reduction [i] based on digital (43, 59, large)-net over F128, using
(58−13, 58, large)-Net in Base 128 — Upper bound on s
There is no (45, 58, large)-net in base 128, because
- 11 times m-reduction [i] would yield (45, 47, large)-net in base 128, but