Best Known (75−30, 75, s)-Nets in Base 128
(75−30, 75, 1242)-Net over F128 — Constructive and digital
Digital (45, 75, 1242)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (1, 16, 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 (29, 59, 1092)-net over F128, using
- net defined by OOA [i] based on linear OOA(12859, 1092, F128, 30, 30) (dual of [(1092, 30), 32701, 31]-NRT-code), using
- OA 15-folding and stacking [i] based on linear OA(12859, 16380, F128, 30) (dual of [16380, 16321, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(12859, 16384, F128, 30) (dual of [16384, 16325, 31]-code), using
- an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- discarding factors / shortening the dual code based on linear OA(12859, 16384, F128, 30) (dual of [16384, 16325, 31]-code), using
- OA 15-folding and stacking [i] based on linear OA(12859, 16380, F128, 30) (dual of [16380, 16321, 31]-code), using
- net defined by OOA [i] based on linear OOA(12859, 1092, F128, 30, 30) (dual of [(1092, 30), 32701, 31]-NRT-code), using
- digital (1, 16, 150)-net over F128, using
(75−30, 75, 4370)-Net in Base 128 — Constructive
(45, 75, 4370)-net in base 128, using
- 1 times m-reduction [i] based on (45, 76, 4370)-net in base 128, using
- net defined by OOA [i] based on OOA(12876, 4370, S128, 31, 31), using
- OOA 15-folding and stacking with additional row [i] based on OA(12876, 65551, S128, 31), using
- discarding factors based on OA(12876, 65554, S128, 31), using
- discarding parts of the base [i] based on linear OA(25666, 65554, F256, 31) (dual of [65554, 65488, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- linear OA(25661, 65537, F256, 31) (dual of [65537, 65476, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(25649, 65537, F256, 25) (dual of [65537, 65488, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(2565, 17, F256, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,256)), using
- discarding factors / shortening the dual code based on linear OA(2565, 256, F256, 5) (dual of [256, 251, 6]-code or 256-arc in PG(4,256)), using
- Reed–Solomon code RS(251,256) [i]
- discarding factors / shortening the dual code based on linear OA(2565, 256, F256, 5) (dual of [256, 251, 6]-code or 256-arc in PG(4,256)), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- discarding parts of the base [i] based on linear OA(25666, 65554, F256, 31) (dual of [65554, 65488, 32]-code), using
- discarding factors based on OA(12876, 65554, S128, 31), using
- OOA 15-folding and stacking with additional row [i] based on OA(12876, 65551, S128, 31), using
- net defined by OOA [i] based on OOA(12876, 4370, S128, 31, 31), using
(75−30, 75, 25897)-Net over F128 — Digital
Digital (45, 75, 25897)-net over F128, using
(75−30, 75, large)-Net in Base 128 — Upper bound on s
There is no (45, 75, large)-net in base 128, because
- 28 times m-reduction [i] would yield (45, 47, large)-net in base 128, but