Best Known (68, 68+18, s)-Nets in Base 16
(68, 68+18, 116511)-Net over F16 — Constructive and digital
Digital (68, 86, 116511)-net over F16, using
- net defined by OOA [i] based on linear OOA(1686, 116511, F16, 18, 18) (dual of [(116511, 18), 2097112, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(1686, 1048599, F16, 18) (dual of [1048599, 1048513, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(1686, 1048600, F16, 18) (dual of [1048600, 1048514, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- linear OA(1681, 1048576, F16, 18) (dual of [1048576, 1048495, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(1661, 1048576, F16, 13) (dual of [1048576, 1048515, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(165, 24, F16, 4) (dual of [24, 19, 5]-code), using
- extended algebraic-geometric code AGe(F,19P) [i] based on function field F/F16 with g(F) = 1 and N(F) ≥ 24, using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(1686, 1048600, F16, 18) (dual of [1048600, 1048514, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(1686, 1048599, F16, 18) (dual of [1048599, 1048513, 19]-code), using
(68, 68+18, 1048601)-Net over F16 — Digital
Digital (68, 86, 1048601)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1686, 1048601, F16, 18) (dual of [1048601, 1048515, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- linear OA(1681, 1048576, F16, 18) (dual of [1048576, 1048495, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(1661, 1048576, F16, 13) (dual of [1048576, 1048515, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(165, 25, F16, 4) (dual of [25, 20, 5]-code), using
- extended algebraic-geometric code AGe(F,20P) [i] based on function field F/F16 with g(F) = 1 and N(F) ≥ 25, using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
(68, 68+18, large)-Net in Base 16 — Upper bound on s
There is no (68, 86, large)-net in base 16, because
- 16 times m-reduction [i] would yield (68, 70, large)-net in base 16, but