Best Known (18, 18+18, s)-Nets in Base 128
(18, 18+18, 1821)-Net over F128 — Constructive and digital
Digital (18, 36, 1821)-net over F128, using
- net defined by OOA [i] based on linear OOA(12836, 1821, F128, 18, 18) (dual of [(1821, 18), 32742, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(12836, 16389, F128, 18) (dual of [16389, 16353, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(15) [i] based on
- linear OA(12835, 16384, F128, 18) (dual of [16384, 16349, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(1281, 5, F128, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(17) ⊂ Ce(15) [i] based on
- OA 9-folding and stacking [i] based on linear OA(12836, 16389, F128, 18) (dual of [16389, 16353, 19]-code), using
(18, 18+18, 4406)-Net over F128 — Digital
Digital (18, 36, 4406)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12836, 4406, F128, 3, 18) (dual of [(4406, 3), 13182, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12836, 5463, F128, 3, 18) (dual of [(5463, 3), 16353, 19]-NRT-code), using
- OOA 3-folding [i] based on linear OA(12836, 16389, F128, 18) (dual of [16389, 16353, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(15) [i] based on
- linear OA(12835, 16384, F128, 18) (dual of [16384, 16349, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(1281, 5, F128, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(17) ⊂ Ce(15) [i] based on
- OOA 3-folding [i] based on linear OA(12836, 16389, F128, 18) (dual of [16389, 16353, 19]-code), using
- discarding factors / shortening the dual code based on linear OOA(12836, 5463, F128, 3, 18) (dual of [(5463, 3), 16353, 19]-NRT-code), using
(18, 18+18, large)-Net in Base 128 — Upper bound on s
There is no (18, 36, large)-net in base 128, because
- 16 times m-reduction [i] would yield (18, 20, large)-net in base 128, but