Best Known (42, 74, s)-Nets in Base 128
(42, 74, 1026)-Net over F128 — Constructive and digital
Digital (42, 74, 1026)-net over F128, using
- 1281 times duplication [i] based on digital (41, 73, 1026)-net over F128, using
- net defined by OOA [i] based on linear OOA(12873, 1026, F128, 32, 32) (dual of [(1026, 32), 32759, 33]-NRT-code), using
- OA 16-folding and stacking [i] based on linear OA(12873, 16416, F128, 32) (dual of [16416, 16343, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(20) [i] based on
- linear OA(12863, 16384, F128, 32) (dual of [16384, 16321, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(12841, 16384, F128, 21) (dual of [16384, 16343, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(12810, 32, F128, 10) (dual of [32, 22, 11]-code or 32-arc in PG(9,128)), using
- discarding factors / shortening the dual code based on linear OA(12810, 128, F128, 10) (dual of [128, 118, 11]-code or 128-arc in PG(9,128)), using
- Reed–Solomon code RS(118,128) [i]
- discarding factors / shortening the dual code based on linear OA(12810, 128, F128, 10) (dual of [128, 118, 11]-code or 128-arc in PG(9,128)), using
- construction X applied to Ce(31) ⊂ Ce(20) [i] based on
- OA 16-folding and stacking [i] based on linear OA(12873, 16416, F128, 32) (dual of [16416, 16343, 33]-code), using
- net defined by OOA [i] based on linear OOA(12873, 1026, F128, 32, 32) (dual of [(1026, 32), 32759, 33]-NRT-code), using
(42, 74, 4096)-Net in Base 128 — Constructive
(42, 74, 4096)-net in base 128, using
- 1 times m-reduction [i] based on (42, 75, 4096)-net in base 128, using
- net defined by OOA [i] based on OOA(12875, 4096, S128, 33, 33), using
- OOA 16-folding and stacking with additional row [i] based on OA(12875, 65537, S128, 33), using
- discarding factors based on OA(12875, 65538, S128, 33), using
- discarding parts of the base [i] based on linear OA(25665, 65538, F256, 33) (dual of [65538, 65473, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(31) [i] based on
- 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(25663, 65536, F256, 32) (dual of [65536, 65473, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [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(32) ⊂ Ce(31) [i] based on
- discarding parts of the base [i] based on linear OA(25665, 65538, F256, 33) (dual of [65538, 65473, 34]-code), using
- discarding factors based on OA(12875, 65538, S128, 33), using
- OOA 16-folding and stacking with additional row [i] based on OA(12875, 65537, S128, 33), using
- net defined by OOA [i] based on OOA(12875, 4096, S128, 33, 33), using
(42, 74, 12707)-Net over F128 — Digital
Digital (42, 74, 12707)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12874, 12707, F128, 32) (dual of [12707, 12633, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(12874, 16419, F128, 32) (dual of [16419, 16345, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(19) [i] based on
- linear OA(12863, 16384, F128, 32) (dual of [16384, 16321, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(12839, 16384, F128, 20) (dual of [16384, 16345, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(31) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(12874, 16419, F128, 32) (dual of [16419, 16345, 33]-code), using
(42, 74, large)-Net in Base 128 — Upper bound on s
There is no (42, 74, large)-net in base 128, because
- 30 times m-reduction [i] would yield (42, 44, large)-net in base 128, but