Best Known (73, 73+19, s)-Nets in Base 16
(73, 73+19, 116511)-Net over F16 — Constructive and digital
Digital (73, 92, 116511)-net over F16, using
- 161 times duplication [i] based on digital (72, 91, 116511)-net over F16, using
- net defined by OOA [i] based on linear OOA(1691, 116511, F16, 19, 19) (dual of [(116511, 19), 2213618, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(1691, 1048600, F16, 19) (dual of [1048600, 1048509, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(1686, 1048576, F16, 19) (dual of [1048576, 1048490, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(1666, 1048576, F16, 14) (dual of [1048576, 1048510, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(18) ⊂ Ce(13) [i] based on
- OOA 9-folding and stacking with additional row [i] based on linear OA(1691, 1048600, F16, 19) (dual of [1048600, 1048509, 20]-code), using
- net defined by OOA [i] based on linear OOA(1691, 116511, F16, 19, 19) (dual of [(116511, 19), 2213618, 20]-NRT-code), using
(73, 73+19, 1048603)-Net over F16 — Digital
Digital (73, 92, 1048603)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1692, 1048603, F16, 19) (dual of [1048603, 1048511, 20]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(1691, 1048601, F16, 19) (dual of [1048601, 1048510, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(1686, 1048576, F16, 19) (dual of [1048576, 1048490, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(1666, 1048576, F16, 14) (dual of [1048576, 1048510, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(18) ⊂ Ce(13) [i] based on
- linear OA(1691, 1048602, F16, 18) (dual of [1048602, 1048511, 19]-code), using Gilbert–Varšamov bound and bm = 1691 > Vbs−1(k−1) = 620585 436471 805881 357826 592673 724948 992655 147825 000920 398635 267056 119804 784634 989617 129628 812416 444544 109016 [i]
- linear OA(160, 1, F16, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(1691, 1048601, F16, 19) (dual of [1048601, 1048510, 20]-code), using
- construction X with Varšamov bound [i] based on
(73, 73+19, large)-Net in Base 16 — Upper bound on s
There is no (73, 92, large)-net in base 16, because
- 17 times m-reduction [i] would yield (73, 75, large)-net in base 16, but