Best Known (65, 65+17, s)-Nets in Base 16
(65, 65+17, 131075)-Net over F16 — Constructive and digital
Digital (65, 82, 131075)-net over F16, using
- net defined by OOA [i] based on linear OOA(1682, 131075, F16, 17, 17) (dual of [(131075, 17), 2228193, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(1682, 1048601, F16, 17) (dual of [1048601, 1048519, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(1682, 1048602, F16, 17) (dual of [1048602, 1048520, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(11) [i] based on
- linear OA(1676, 1048576, F16, 17) (dual of [1048576, 1048500, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(1656, 1048576, F16, 12) (dual of [1048576, 1048520, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(166, 26, F16, 4) (dual of [26, 20, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(166, 240, F16, 4) (dual of [240, 234, 5]-code), using
- 1 times truncation [i] based on linear OA(167, 241, F16, 5) (dual of [241, 234, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(166, 240, F16, 4) (dual of [240, 234, 5]-code), using
- construction X applied to Ce(16) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(1682, 1048602, F16, 17) (dual of [1048602, 1048520, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(1682, 1048601, F16, 17) (dual of [1048601, 1048519, 18]-code), using
(65, 65+17, 1048603)-Net over F16 — Digital
Digital (65, 82, 1048603)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1682, 1048603, F16, 17) (dual of [1048603, 1048521, 18]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(1681, 1048601, F16, 17) (dual of [1048601, 1048520, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(11) [i] based on
- linear OA(1676, 1048576, F16, 17) (dual of [1048576, 1048500, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(1656, 1048576, F16, 12) (dual of [1048576, 1048520, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(16) ⊂ Ce(11) [i] based on
- linear OA(1681, 1048602, F16, 16) (dual of [1048602, 1048521, 17]-code), using Gilbert–Varšamov bound and bm = 1681 > Vbs−1(k−1) = 682307 608916 241845 176724 097322 594486 024916 893402 078420 603010 808821 785212 013268 341951 356653 484016 [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(1681, 1048601, F16, 17) (dual of [1048601, 1048520, 18]-code), using
- construction X with Varšamov bound [i] based on
(65, 65+17, large)-Net in Base 16 — Upper bound on s
There is no (65, 82, large)-net in base 16, because
- 15 times m-reduction [i] would yield (65, 67, large)-net in base 16, but