Best Known (78−34, 78, s)-Nets in Base 128
(78−34, 78, 965)-Net over F128 — Constructive and digital
Digital (44, 78, 965)-net over F128, using
- 1282 times duplication [i] based on digital (42, 76, 965)-net over F128, using
- t-expansion [i] based on digital (41, 76, 965)-net over F128, using
- net defined by OOA [i] based on linear OOA(12876, 965, F128, 35, 35) (dual of [(965, 35), 33699, 36]-NRT-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(12876, 16406, F128, 35) (dual of [16406, 16330, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(12876, 16408, F128, 35) (dual of [16408, 16332, 36]-code), using
- construction X applied to C([0,17]) ⊂ C([0,13]) [i] based on
- linear OA(12869, 16385, F128, 35) (dual of [16385, 16316, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,17], and minimum distance d ≥ |{−17,−16,…,17}|+1 = 36 (BCH-bound) [i]
- linear OA(12853, 16385, F128, 27) (dual of [16385, 16332, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(1287, 23, F128, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,128)), using
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- Reed–Solomon code RS(121,128) [i]
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- construction X applied to C([0,17]) ⊂ C([0,13]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12876, 16408, F128, 35) (dual of [16408, 16332, 36]-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(12876, 16406, F128, 35) (dual of [16406, 16330, 36]-code), using
- net defined by OOA [i] based on linear OOA(12876, 965, F128, 35, 35) (dual of [(965, 35), 33699, 36]-NRT-code), using
- t-expansion [i] based on digital (41, 76, 965)-net over F128, using
(78−34, 78, 3855)-Net in Base 128 — Constructive
(44, 78, 3855)-net in base 128, using
- 1281 times duplication [i] based on (43, 77, 3855)-net in base 128, using
- net defined by OOA [i] based on OOA(12877, 3855, S128, 34, 34), using
- OA 17-folding and stacking [i] based on OA(12877, 65535, S128, 34), using
- discarding factors based on OA(12877, 65538, S128, 34), using
- discarding parts of the base [i] based on linear OA(25667, 65538, F256, 34) (dual of [65538, 65471, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(32) [i] based on
- linear OA(25667, 65536, F256, 34) (dual of [65536, 65469, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(25665, 65536, F256, 33) (dual of [65536, 65471, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(33) ⊂ Ce(32) [i] based on
- discarding parts of the base [i] based on linear OA(25667, 65538, F256, 34) (dual of [65538, 65471, 35]-code), using
- discarding factors based on OA(12877, 65538, S128, 34), using
- OA 17-folding and stacking [i] based on OA(12877, 65535, S128, 34), using
- net defined by OOA [i] based on OOA(12877, 3855, S128, 34, 34), using
(78−34, 78, 11830)-Net over F128 — Digital
Digital (44, 78, 11830)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12878, 11830, F128, 34) (dual of [11830, 11752, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(12878, 16419, F128, 34) (dual of [16419, 16341, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(21) [i] based on
- linear OA(12867, 16384, F128, 34) (dual of [16384, 16317, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(12843, 16384, F128, 22) (dual of [16384, 16341, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(12811, 35, F128, 11) (dual of [35, 24, 12]-code or 35-arc in PG(10,128)), using
- discarding factors / shortening the dual code based on linear OA(12811, 128, F128, 11) (dual of [128, 117, 12]-code or 128-arc in PG(10,128)), using
- Reed–Solomon code RS(117,128) [i]
- discarding factors / shortening the dual code based on linear OA(12811, 128, F128, 11) (dual of [128, 117, 12]-code or 128-arc in PG(10,128)), using
- construction X applied to Ce(33) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(12878, 16419, F128, 34) (dual of [16419, 16341, 35]-code), using
(78−34, 78, large)-Net in Base 128 — Upper bound on s
There is no (44, 78, large)-net in base 128, because
- 32 times m-reduction [i] would yield (44, 46, large)-net in base 128, but