Best Known (42, 42+30, s)-Nets in Base 128
(42, 42+30, 1095)-Net over F128 — Constructive and digital
Digital (42, 72, 1095)-net over F128, using
- net defined by OOA [i] based on linear OOA(12872, 1095, F128, 30, 30) (dual of [(1095, 30), 32778, 31]-NRT-code), using
- OA 15-folding and stacking [i] based on linear OA(12872, 16425, F128, 30) (dual of [16425, 16353, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(15) [i] 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]
- linear OA(12831, 16384, F128, 16) (dual of [16384, 16353, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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 Ce(29) ⊂ Ce(15) [i] based on
- OA 15-folding and stacking [i] based on linear OA(12872, 16425, F128, 30) (dual of [16425, 16353, 31]-code), using
(42, 42+30, 4370)-Net in Base 128 — Constructive
(42, 72, 4370)-net in base 128, using
- base change [i] based on digital (33, 63, 4370)-net over F256, using
- net defined by OOA [i] based on linear OOA(25663, 4370, F256, 30, 30) (dual of [(4370, 30), 131037, 31]-NRT-code), using
- OA 15-folding and stacking [i] based on linear OA(25663, 65550, F256, 30) (dual of [65550, 65487, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(24) [i] based on
- linear OA(25659, 65536, F256, 30) (dual of [65536, 65477, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(25649, 65536, F256, 25) (dual of [65536, 65487, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(2564, 14, F256, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,256)), using
- discarding factors / shortening the dual code based on linear OA(2564, 256, F256, 4) (dual of [256, 252, 5]-code or 256-arc in PG(3,256)), using
- Reed–Solomon code RS(252,256) [i]
- discarding factors / shortening the dual code based on linear OA(2564, 256, F256, 4) (dual of [256, 252, 5]-code or 256-arc in PG(3,256)), using
- construction X applied to Ce(29) ⊂ Ce(24) [i] based on
- OA 15-folding and stacking [i] based on linear OA(25663, 65550, F256, 30) (dual of [65550, 65487, 31]-code), using
- net defined by OOA [i] based on linear OOA(25663, 4370, F256, 30, 30) (dual of [(4370, 30), 131037, 31]-NRT-code), using
(42, 42+30, 16425)-Net over F128 — Digital
Digital (42, 72, 16425)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12872, 16425, F128, 30) (dual of [16425, 16353, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(15) [i] 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]
- linear OA(12831, 16384, F128, 16) (dual of [16384, 16353, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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 Ce(29) ⊂ Ce(15) [i] based on
(42, 42+30, large)-Net in Base 128 — Upper bound on s
There is no (42, 72, large)-net in base 128, because
- 28 times m-reduction [i] would yield (42, 44, large)-net in base 128, but