Best Known (69−15, 69, s)-Nets in Base 128
(69−15, 69, 1198758)-Net over F128 — Constructive and digital
Digital (54, 69, 1198758)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (5, 12, 387)-net over F128, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 2, 129)-net over F128, using
- 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, 7, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128 (see above)
- generalized (u, u+v)-construction [i] based on
- digital (42, 57, 1198371)-net over F128, using
- net defined by OOA [i] based on linear OOA(12857, 1198371, F128, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(12857, 8388598, F128, 15) (dual of [8388598, 8388541, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(12857, large, F128, 15) (dual of [large, large−57, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 15790321 | 1288−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(12857, large, F128, 15) (dual of [large, large−57, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(12857, 8388598, F128, 15) (dual of [8388598, 8388541, 16]-code), using
- net defined by OOA [i] based on linear OOA(12857, 1198371, F128, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
- digital (5, 12, 387)-net over F128, using
(69−15, 69, 1897422)-Net in Base 128 — Constructive
(54, 69, 1897422)-net in base 128, using
- (u, u+v)-construction [i] based on
- digital (12, 19, 699051)-net over F128, using
- net defined by OOA [i] based on linear OOA(12819, 699051, F128, 7, 7) (dual of [(699051, 7), 4893338, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(12819, 2097154, F128, 7) (dual of [2097154, 2097135, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(12819, 2097155, F128, 7) (dual of [2097155, 2097136, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(12819, 2097152, F128, 7) (dual of [2097152, 2097133, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(12816, 2097152, F128, 6) (dual of [2097152, 2097136, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(12819, 2097155, F128, 7) (dual of [2097155, 2097136, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(12819, 2097154, F128, 7) (dual of [2097154, 2097135, 8]-code), using
- net defined by OOA [i] based on linear OOA(12819, 699051, F128, 7, 7) (dual of [(699051, 7), 4893338, 8]-NRT-code), using
- (35, 50, 1198371)-net in base 128, using
- net defined by OOA [i] based on OOA(12850, 1198371, S128, 15, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(12850, 8388598, S128, 15), using
- discarding factors based on OA(12850, large, S128, 15), using
- discarding parts of the base [i] based on linear OA(25643, large, F256, 15) (dual of [large, large−43, 16]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding parts of the base [i] based on linear OA(25643, large, F256, 15) (dual of [large, large−43, 16]-code), using
- discarding factors based on OA(12850, large, S128, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(12850, 8388598, S128, 15), using
- net defined by OOA [i] based on OOA(12850, 1198371, S128, 15, 15), using
- digital (12, 19, 699051)-net over F128, using
(69−15, 69, large)-Net over F128 — Digital
Digital (54, 69, large)-net over F128, using
- 5 times m-reduction [i] based on digital (54, 74, large)-net over F128, using
(69−15, 69, large)-Net in Base 128 — Upper bound on s
There is no (54, 69, large)-net in base 128, because
- 13 times m-reduction [i] would yield (54, 56, large)-net in base 128, but