Best Known (67−15, 67, s)-Nets in Base 128
(67−15, 67, 1198629)-Net over F128 — Constructive and digital
Digital (52, 67, 1198629)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (3, 10, 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, 7, 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 (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 (3, 10, 258)-net over F128, using
(67−15, 67, 1220218)-Net in Base 128 — Constructive
(52, 67, 1220218)-net in base 128, using
- 1281 times duplication [i] based on (51, 66, 1220218)-net in base 128, using
- (u, u+v)-construction [i] based on
- (9, 16, 21847)-net in base 128, using
- base change [i] based on digital (7, 14, 21847)-net over F256, using
- net defined by OOA [i] based on linear OOA(25614, 21847, F256, 7, 7) (dual of [(21847, 7), 152915, 8]-NRT-code), using
- appending kth column [i] based on linear OOA(25614, 21847, F256, 6, 7) (dual of [(21847, 6), 131068, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(25614, 65542, F256, 7) (dual of [65542, 65528, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(25613, 65537, F256, 7) (dual of [65537, 65524, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(2569, 65537, F256, 5) (dual of [65537, 65528, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(2561, 5, F256, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, s, F256, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(25614, 65542, F256, 7) (dual of [65542, 65528, 8]-code), using
- appending kth column [i] based on linear OOA(25614, 21847, F256, 6, 7) (dual of [(21847, 6), 131068, 8]-NRT-code), using
- net defined by OOA [i] based on linear OOA(25614, 21847, F256, 7, 7) (dual of [(21847, 7), 152915, 8]-NRT-code), using
- base change [i] based on digital (7, 14, 21847)-net over F256, 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
- (9, 16, 21847)-net in base 128, using
- (u, u+v)-construction [i] based on
(67−15, 67, large)-Net over F128 — Digital
Digital (52, 67, large)-net over F128, using
- t-expansion [i] based on digital (51, 67, large)-net over F128, using
- 3 times m-reduction [i] based on digital (51, 70, large)-net over F128, using
(67−15, 67, large)-Net in Base 128 — Upper bound on s
There is no (52, 67, large)-net in base 128, because
- 13 times m-reduction [i] would yield (52, 54, large)-net in base 128, but