Best Known (75−17, 75, s)-Nets in Base 128
(75−17, 75, 1048725)-Net over F128 — Constructive and digital
Digital (58, 75, 1048725)-net over F128, using
- 1281 times duplication [i] based on digital (57, 74, 1048725)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (1, 9, 150)-net over F128, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 1 and N(F) ≥ 150, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- digital (48, 65, 1048575)-net over F128, using
- net defined by OOA [i] based on linear OOA(12865, 1048575, F128, 17, 17) (dual of [(1048575, 17), 17825710, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(12865, 8388601, F128, 17) (dual of [8388601, 8388536, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(12865, large, F128, 17) (dual of [large, large−65, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 15790321 | 1288−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(12865, large, F128, 17) (dual of [large, large−65, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(12865, 8388601, F128, 17) (dual of [8388601, 8388536, 18]-code), using
- net defined by OOA [i] based on linear OOA(12865, 1048575, F128, 17, 17) (dual of [(1048575, 17), 17825710, 18]-NRT-code), using
- digital (1, 9, 150)-net over F128, using
- (u, u+v)-construction [i] based on
(75−17, 75, 1064960)-Net in Base 128 — Constructive
(58, 75, 1064960)-net in base 128, using
- (u, u+v)-construction [i] based on
- (11, 19, 16385)-net in base 128, using
- net defined by OOA [i] based on OOA(12819, 16385, S128, 8, 8), using
- OA 4-folding and stacking [i] based on OA(12819, 65540, S128, 8), using
- discarding factors based on OA(12819, 65541, S128, 8), using
- discarding parts of the base [i] based on linear OA(25616, 65541, F256, 8) (dual of [65541, 65525, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(25615, 65536, F256, 8) (dual of [65536, 65521, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- 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(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 Ce(7) ⊂ Ce(5) [i] based on
- discarding parts of the base [i] based on linear OA(25616, 65541, F256, 8) (dual of [65541, 65525, 9]-code), using
- discarding factors based on OA(12819, 65541, S128, 8), using
- OA 4-folding and stacking [i] based on OA(12819, 65540, S128, 8), using
- net defined by OOA [i] based on OOA(12819, 16385, S128, 8, 8), using
- (39, 56, 1048575)-net in base 128, using
- base change [i] based on digital (32, 49, 1048575)-net over F256, using
- net defined by OOA [i] based on linear OOA(25649, 1048575, F256, 17, 17) (dual of [(1048575, 17), 17825726, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(25649, 8388601, F256, 17) (dual of [8388601, 8388552, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(25649, large, F256, 17) (dual of [large, large−49, 18]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(25649, large, F256, 17) (dual of [large, large−49, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(25649, 8388601, F256, 17) (dual of [8388601, 8388552, 18]-code), using
- net defined by OOA [i] based on linear OOA(25649, 1048575, F256, 17, 17) (dual of [(1048575, 17), 17825726, 18]-NRT-code), using
- base change [i] based on digital (32, 49, 1048575)-net over F256, using
- (11, 19, 16385)-net in base 128, using
(75−17, 75, large)-Net over F128 — Digital
Digital (58, 75, large)-net over F128, using
- t-expansion [i] based on digital (56, 75, large)-net over F128, using
- 2 times m-reduction [i] based on digital (56, 77, large)-net over F128, using
(75−17, 75, large)-Net in Base 128 — Upper bound on s
There is no (58, 75, large)-net in base 128, because
- 15 times m-reduction [i] would yield (58, 60, large)-net in base 128, but