Best Known (34−16, 34, s)-Nets in Base 128
(34−16, 34, 2049)-Net over F128 — Constructive and digital
Digital (18, 34, 2049)-net over F128, using
- 1281 times duplication [i] based on digital (17, 33, 2049)-net over F128, using
- net defined by OOA [i] based on linear OOA(12833, 2049, F128, 16, 16) (dual of [(2049, 16), 32751, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(12833, 16392, F128, 16) (dual of [16392, 16359, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- 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(12825, 16384, F128, 13) (dual of [16384, 16359, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(1282, 8, F128, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,128)), using
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- Reed–Solomon code RS(126,128) [i]
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- OA 8-folding and stacking [i] based on linear OA(12833, 16392, F128, 16) (dual of [16392, 16359, 17]-code), using
- net defined by OOA [i] based on linear OOA(12833, 2049, F128, 16, 16) (dual of [(2049, 16), 32751, 17]-NRT-code), using
(34−16, 34, 6858)-Net over F128 — Digital
Digital (18, 34, 6858)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12834, 6858, F128, 2, 16) (dual of [(6858, 2), 13682, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12834, 8197, F128, 2, 16) (dual of [(8197, 2), 16360, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12834, 16394, F128, 16) (dual of [16394, 16360, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(12834, 16395, F128, 16) (dual of [16395, 16361, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- 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(12823, 16384, F128, 12) (dual of [16384, 16361, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(1283, 11, F128, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,128) or 11-cap in PG(2,128)), using
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- Reed–Solomon code RS(125,128) [i]
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(12834, 16395, F128, 16) (dual of [16395, 16361, 17]-code), using
- OOA 2-folding [i] based on linear OA(12834, 16394, F128, 16) (dual of [16394, 16360, 17]-code), using
- discarding factors / shortening the dual code based on linear OOA(12834, 8197, F128, 2, 16) (dual of [(8197, 2), 16360, 17]-NRT-code), using
(34−16, 34, large)-Net in Base 128 — Upper bound on s
There is no (18, 34, large)-net in base 128, because
- 14 times m-reduction [i] would yield (18, 20, large)-net in base 128, but