Best Known (75−31, 75, s)-Nets in Base 128
(75−31, 75, 1095)-Net over F128 — Constructive and digital
Digital (44, 75, 1095)-net over F128, using
- 1281 times duplication [i] based on digital (43, 74, 1095)-net over F128, using
- net defined by OOA [i] based on linear OOA(12874, 1095, F128, 31, 31) (dual of [(1095, 31), 33871, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(12874, 16426, F128, 31) (dual of [16426, 16352, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,8]) [i] based on
- linear OA(12861, 16385, F128, 31) (dual of [16385, 16324, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(12833, 16385, F128, 17) (dual of [16385, 16352, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(12813, 41, F128, 13) (dual of [41, 28, 14]-code or 41-arc in PG(12,128)), using
- discarding factors / shortening the dual code based on linear OA(12813, 128, F128, 13) (dual of [128, 115, 14]-code or 128-arc in PG(12,128)), using
- Reed–Solomon code RS(115,128) [i]
- discarding factors / shortening the dual code based on linear OA(12813, 128, F128, 13) (dual of [128, 115, 14]-code or 128-arc in PG(12,128)), using
- construction X applied to C([0,15]) ⊂ C([0,8]) [i] based on
- OOA 15-folding and stacking with additional row [i] based on linear OA(12874, 16426, F128, 31) (dual of [16426, 16352, 32]-code), using
- net defined by OOA [i] based on linear OOA(12874, 1095, F128, 31, 31) (dual of [(1095, 31), 33871, 32]-NRT-code), using
(75−31, 75, 4369)-Net in Base 128 — Constructive
(44, 75, 4369)-net in base 128, using
- 1283 times duplication [i] based on (41, 72, 4369)-net in base 128, using
- base change [i] based on digital (32, 63, 4369)-net over F256, using
- 2562 times duplication [i] based on digital (30, 61, 4369)-net over F256, using
- net defined by OOA [i] based on linear OOA(25661, 4369, F256, 31, 31) (dual of [(4369, 31), 135378, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(25661, 65536, F256, 31) (dual of [65536, 65475, 32]-code), using
- an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- OOA 15-folding and stacking with additional row [i] based on linear OA(25661, 65536, F256, 31) (dual of [65536, 65475, 32]-code), using
- net defined by OOA [i] based on linear OOA(25661, 4369, F256, 31, 31) (dual of [(4369, 31), 135378, 32]-NRT-code), using
- 2562 times duplication [i] based on digital (30, 61, 4369)-net over F256, using
- base change [i] based on digital (32, 63, 4369)-net over F256, using
(75−31, 75, 17595)-Net over F128 — Digital
Digital (44, 75, 17595)-net over F128, using
(75−31, 75, large)-Net in Base 128 — Upper bound on s
There is no (44, 75, large)-net in base 128, because
- 29 times m-reduction [i] would yield (44, 46, large)-net in base 128, but